RADU GROUPS ACTING ON TREES ARE CCR

Abstract

We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting $2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia’s CCR conjecture on trees for $(d_0,d_1)$-semi-regular trees such that $d_0,d_1\in \Theta $, where $\Theta $ is an asymptotically dense set of positive integers.

1 Introduction

In this document, topological groups are second-countable, locally compact groups are Hausdorff and the word ‘representation’ stands for a strongly continuous unitary representation on a separable complex Hilbert space. A locally compact group G is called CCR if the operator $\pi (f)$ is compact for all irreducible representations $\pi $ of G and all $f\in L^{1}(G)$ . For totally disconnected locally compact groups, this property is equivalent to the requirement that every irreducible representation of G is admissible; see [Neb99]. We recall that an irreducible representation $\pi $ of a totally disconnected locally compact group G is admissible if for every compact open subgroup $K \leq G$ , the space $\mathcal {H}_{\pi }^K$ of K-invariant vectors is finite dimensional. A very important property of CCR groups is that they are type I groups [BdlH20, Definition 6.E.7. and Proposition 6.E.11]. Loosely speaking, type I groups are the locally compact groups all of whose representations can be written as unique direct integrals of irreducible representations, thus reducing the study of arbitrary representations to considerations of irreducible representations. Concerning groups of automorphisms of trees, Nebbia’s work highlighted surprising relations between the action on the boundary and the regularity of representation theory. To be more precise, he showed in [Neb99] that any closed unimodular CCR vertex-transitive subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ of the group of automorphisms of a regular tree T necessarily acts transitively on the boundary $\partial T$ . Further progress going in that direction was recently achieved by Houdayer and Raum [HR19] and, at a higher level of generality, by Caprace et al. [CKM23]. Among other things, they showed that a closed nonamenable type I subgroup acting minimally on a locally finite tree T acts $2$ -transitively on the boundary $\partial T$ [CKM23, Corollary D]. Going in the other direction, Nebbia conjectured in [Neb99] that any closed subgroup of automorphisms of a regular tree acting transitively on the boundary is CCR. His conjecture naturally extends to the case of semi-regular trees.

Conjecture (CCR conjecture on trees [Neb99]).

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 3$ and let $G\leq \operatorname {\mathrm {Aut}}(T)$ be a closed subgroup acting transitively on the boundary of T. Then G is CCR.

We recall from [BM00, Lemma 3.1.1] that, for a locally finite tree T, closed subgroups $G\leq \operatorname {\mathrm {Aut}}(T)$ are noncompact and act transitively on the boundary $\partial T$ if and only if they act $2$ -transitively on $\partial T$ . Furthermore, the existence of such a group implies that the tree is semi-regular. In particular, since compact groups are automatically CCR, the hypothesis of semi-regularity is not restrictive in the conjecture.

One of the first pieces of evidence supporting the conjecture was provided by Bernstein and Harish-Chandra’s works. Among other things, they proved that rank one semi-simple algebraic groups over local-fields are uniformly admissible [Ber74, HC70]. We recall that a totally disconnected locally compact group G is uniformly admissible if for every compact open subgroup K, there exists a positive integer $k_K$ such that ${\dim (\mathcal {H}_{\pi }^K)<k_K}$ for all irreducible representations $\pi $ of G. In particular, uniformly admissible groups are CCR. Concerning nonlinear groups, the conjecture was supported by the complete classification of the irreducible representations of the full group of automorphisms of a semi-regular tree and more generally of closed subgroups acting transitively on the boundary and satisfying the Tits independence property [Ama03, Cho94, FTN91, Ol’77, Ol’82] (these classifications lead to the conclusion that they are uniformly admissible [Cio15]).

Our paper concerns closed subgroups acting $2$ -transitively on the boundary $\partial T$ and whose local action at each vertex v contains the alternating group of corresponding degree. We recall that for each vertex $v\in V(T)$ , the stabiliser $\mathrm {Fix}_G(v)$ of v acts on the set $E(v)$ of edges containing v. The image of $\mathrm {Fix}_G(v)$ in $\operatorname {\mathrm {Sym}}(E(v))$ for this natural projection map is called the local action of G at v and we denote this group by $\underline {G}(v)$ . When the degree of each vertex is greater than $6$ , these groups of automorphisms of trees have been extensively studied and classified by Radu in [Rad17]. For this reason, we call them Radu groups. It is not hard to realise that these groups are type I. Indeed, each Radu group G contains a cocompact subgroup H that is conjugate in $\operatorname {\mathrm {Aut}}(T)$ to the semi-regular version of the universal group of Burger–Mozes $\operatorname {\mathrm {Alt}}_{(i)}(T)^+$ ; see [Rad17, page 4]. Since H is both open and cocompact in G, [Kal70, Theorem 1] ensures that G is type I if and only if H is type I. However, when the degree of each vertex is greater than $4$ , $\operatorname {\mathrm {Alt}}_{(i)}(T)^+$ acts transitively on the boundary and satisfies the Tits independence property. It follows from [Ama03, Cio15] that H is a type I group, which proves that every Radu group is type I. The purpose of these notes is to go further. Inspired by Ol’shanskii’s work and the recent progress achieved on the abstraction of his framework [Sem23], we give a classification of the irreducible representations of simple Radu groups and deduce a description of the irreducible representations of any Radu group. Among other things, we provide the following contribution to Nebbia’s CCR conjecture.

Theorem A. Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 6$ . Then, Radu groups are uniformly admissible and hence CCR.

To put this result into the perspective of Radu’s paper, we recall that the local action $\underline {G}(v)\leq \operatorname {\mathrm {Sym}}(E(v))$ at every vertex $v\in T$ of a closed subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ that is $2$ -transitive on the boundary is a $2$ -transitive subgroup of $\operatorname {\mathrm {Sym}}(E(v))$ [BM00, Lemma 3.1.1]. However, [Rad17, Proposition B.1 and Corollary B.2] ensure that

$$ \begin{align*} \Theta= \{d \geq 6 \mid \mbox{each finite }\mbox{2-transitive subgroup of }\operatorname{\mathrm{Sym}}(d)\mbox{ contains Alt}(d)\} \end{align*} $$

is asymptotically dense in $\mathbb N$ and its ten smallest elements are $34$ , $35$ , $39$ , $45$ , $46$ , $51$ , $52$ , $55$ , $56$ and $58$ . All together, this implies the following.

Theorem B. Nebbia’s CCR conjecture on trees is confirmed for any $(d_0,d_1)$ -semi- regular tree with $d_0,d_1\in \Theta $ where $\Theta $ is the asymptotically dense subset of $\mathbb N$ defined above.

We now come back to the classification of the irreducible representations of simple Radu groups. We recall that the irreducible representations of a closed automorphism group $G\leq \operatorname {\mathrm {Aut}}(T)$ of a locally finite tree T split into three categories. An irreducible representation $\pi $ of G is called:

• • spherical if there exists a vertex $v\in V(T)$ such that $\pi $ admits a nonzero $\mathrm {Fix}_G(v)$ -invariant vector where $\mathrm {Fix}_G(v)=\{g\in G\mid gv=v\}$ ;

• • special if it is not spherical and there exists an edge $e\in E(T)$ such that $\pi $ admits a nonzero $\mathrm {Fix}_G(e)$ -invariant vector where $\mathrm {Fix}_G(e)=\{g\in G\mid gv=v\,\text { for all } v\in e\}$ is the fixator of the edge e;

• • cuspidal if for every $e \in E(T)$ , $\pi $ does not admit a nonzero $\mathrm {Fix}_G(e)$ -invariant vector.

The spherical and special representations are classified since the end of the 1970s at the level of generality of the conjecture, that is, for any closed noncompact subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ acting transitively on the boundary of the tree; see [Mat77, Ol’77, Ol’82]. Furthermore, we recall that Matsumoto’s work emphasises a strong connection between these kinds of representations and the irreducible representations of Hecke algebras. To be more precise, we recall that a group acting $2$ -transitively on the boundary is either type-preserving or admits a closed type-preserving subgroup of index $2$ acting $2$ -transitively on the boundary. Since [CC15, Corollary 3.6] ensures that every closed type-preserving subgroup G acting $2$ -transitively on the boundary comes from a B–N pair, every spherical and every special representation of G also provide an irreducible representation of the associated Hecke algebra $C_c(B\backslash G/ B)$ of continuous compactly supported B-bi-invariant functions $f:G\rightarrow \mathbb C$ , where ${B=\mathrm {Fix}_G(e)}$ is the pointwise fixator of an edge $e\in E(T)$ . Matsumoto’s works revealed that this correspondence is actually bijective; see [Mat77, Ch. 5, Section 6].

The cuspidal representations, however, are not classified at the level of generality of the conjecture. Nevertheless, a classification of the cuspidal representations was achieved for certain families of groups. Concerning nonlinear groups, Ol’shanskii obtained such a classification for any closed subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ satisfying the Tits independence property by exploiting the independence of the action on the tree to deduce a particular factorisation of the compact open subgroups; see [Ama03, Ol’77]. Our paper takes advantage of the recent abstraction of his framework given by the notion of Ol’shanskii’s factorisation (see [Sem23] or Definition 4.3 below) and the description of Radu groups [Rad17] to obtain a classification for their cuspidal representations. We start by considering a family of groups $G_{(i)}^+(Y_0,Y_1)$ indexed by finite subsets $Y_0,Y_1\subseteq \mathbb N$ ; see Definition 3.3. It is shown in [Rad17] that these groups are abstractly simple when T is a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ and that they exhaust the list of simple Radu groups when $d_0,d_1\geq 6$ . Furthermore, by exploiting the fact that these groups are determined by suitable local conditions, [Sem23] leads to the following result.

Theorem C. Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ . The cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ are in bijective correspondence given by induction with a family of irreducible representations of compact open subgroups. This correspondence is explicitly given by Theorem 4.12.

Among other things, this theorem proves that the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ are square-integrable. In the light of [HC70, Corollary of Theorem 2] and of the classification of spherical and special representations recalled in Section 2, this proves that the groups $G_{(i)}^+(Y_0,Y_1)$ are uniformly admissible, and hence CCR (the details are gathered in Section 5). To put these results into the perspective of Radu’s classification, we recall that every Radu group G belongs to a finite chain ${H_n\geq \cdots \geq H_0}$ with $n\in \{0,1,2,3\}$ such that $H_n=G$ , $\lbrack H_t: H_{t-1} \rbrack =2$ for all t and $H_0$ is conjugate in the group of type-preserving automorphisms $\operatorname {\mathrm {Aut}}(T)^+$ to one of these $G_{(i)}^+(Y_0,Y_1)$ when $d_0,d_1\geq 6$ . However, Mackey’s machinery allows one to describe the irreducible representation of a locally compact G in terms of the irreducible representations of any of its closed subgroups of index $2$ . In particular, when $d_0,d_1\geq 6$ , we obtain a description of the cuspidal representations of any Radu group from the cuspidal representations of the groups $G_{(i)}^+(Y_0,Y_1)$ . We also deduce from Mackey’s machinery that every Radu group is uniformly admissible. The author would like to underline that an application of Ol’shanskii’s framework to Radu groups is already presented in [Sem23, Section 4] since these groups satisfy a generalisation of the Tits independence property (the property $\mathrm {IP} _k$ introduced in [BEW15]). However, except when the property $\mathrm {IP} _k$ is the Tits independence property ( $k=1$ ), the approach adopted in [Sem23] never leads to a classification of all the cuspidal representations.

1.1 Structure of the paper

In Section 2, we recall the classification of spherical and special representations of any closed noncompact subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ acting transitively on the boundary; see [Mat77, Ol’77, Ol’82]. The purpose of Section 3 is to recall Radu’s classification of Radu groups [Rad17] and the definition of $G_{(i)}^+(Y_0,Y_1)$ . In Section 4, we recall the notion of Ol’shanskii’s factorisation developed in [Sem23] and obtain a classification of the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ . The complete classification of the irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ resulting from Sections 2 and 4 is then used in Section 5 to prove uniform admissibility. Finally, the purpose of the Appendix is to recall the details of the correspondence given by Mackey’s machinery between the irreducible representations of a locally compact G and the irreducible representations of any of its closed subgroups of index $2$ . In particular, the Appendix provides a way to obtain the irreducible representations of any Radu group from the irreducible representations of the abstractly simple Radu groups $G_{(i)}^+(Y_0,Y_1)$ . We also deduce from the Appendix that all Radu groups are uniformly admissible when $d_0,d_1 \geq 6$ .

2 Spherical and special representations

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 3$ . We recall that a tree T is called $(d_0,d_1)$ -semi-regular if there exists a bipartition $V(T)=V_0\sqcup V_1$ of T such that every vertex of $V_i$ has degree $d_i$ and every edge of T contains exactly one vertex in each $V_i$ . As explained in Section 1, the irreducible representations of any closed subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ of the group of automorphisms of such a tree split into three categories. These representations are either spherical, special or cuspidal. The purpose of the present section is to recall the classification of spherical and special representations of any closed noncompact subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ acting transitively on the boundary (in particular of any Radu group). This classification is a classical result known since the end of the 70’s and we claim no originality. We refer to [Mat77, Ol’77, FTN91] for details.

The details of this classification are gathered in Theorems 2.2, 2.3, 2.5 below but we start with some preliminaries. Given a locally compact group G and a compact subgroup $K\leq G$ , we say that $(G, K)$ is a Gelfand pair if the convolution algebra $C_c(K \backslash G / K)$ of compactly supported, continuous K-bi-invariant functions on G is commutative. Let $(G,K)$ be a Gelfand pair and let $\mu $ be the left-Haar measure of G renormalised so that $\mu (K)=1$ . A function $\varphi : G\rightarrow \mathbb C$ is called K -spherical if it is a K-bi-invariant continuous function with $\varphi (1_G)=1$ and such that

$$ \begin{align*}\int_{K}\varphi(gkg')\, d \mu(k)=\varphi(g)\varphi(g') \quad \forall g,g'\in G.\end{align*} $$

Theorem 2.1 [Lan85, Ch. IV, Section 3, Theorems $3$ and $9$ ].

Let $(G,K)$ be a Gelfand pair. For every irreducible representation $\pi $ of G we have that $\dim (\mathcal {H}_{\pi }^K)\leq 1$ . Furthermore, there is a bijective correspondence $\pi \rightarrow \varphi _\pi $ with inverse map given by the GNS construction between the equivalence classes of irreducible representations of G with nonzero K-invariant vectors and the K-spherical functions of positive type on G (the function $\varphi _\pi $ is the function $\varphi _{\pi }(g)=\langle \pi (g)\xi ,\xi \rangle $ corresponding to any unit vector $\xi \in \mathcal {H}_{\pi }^K$ ).

We now recall the details of the classification of spherical and special representations for any noncompact closed subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ acting transitively on the boundary of T. We recall that these groups have either one or two orbits of vertices and we treat these cases separately.

Theorem 2.2 [FTN91, Ch. II].

Let T be a d-regular tree, let $v\in V(T)$ and let ${G\leq \operatorname {\mathrm {Aut}}(T)}$ be a closed noncompact subgroup acting transitively on the vertices of T and the boundary $\partial T$ . Then, $(G, \mathrm {Fix}_G(v))$ is a Gelfand pair and every spherical representation of G admits a nonzero $\mathrm {Fix}_G(v)$ -invariant vector. Furthermore, the equivalence classes of spherical representations of G are in bijective correspondence with the interval $\lbrack -1; 1\rbrack $ via the map $\phi _v:\pi \mapsto \varphi _\pi (\tau _v)$ where $\tau _v$ is any element of G such that $d(\tau _v v,v)=1$ and $\varphi _\pi $ is the unique $\mathrm {Fix}_G(v)$ -spherical function of positive type attached with $\pi $ . Under this correspondence, the trivial representation corresponds to $1$ .

The following theorem comes from [Mat77] but is formulated differently for the sake of consistency.

Theorem 2.3 [Mat77, Ch. 5, Section 6].

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 3$ , let $v\in V(T)$ , let $v'$ be any vertex at distance one from v and let ${G\leq \operatorname {\mathrm {Aut}}(T)^+}$ be a closed noncompact subgroup of type-preserving automorphisms acting transitively on the boundary $\partial T$ . Then, there is exactly one spherical representation $\pi _v$ of G with a nonzero $\mathrm {Fix}_G(v)$ -invariant vector but no nonzero $\mathrm {Fix}_G(v')$ -invariant vector. Furthermore, $(G, \mathrm {Fix}_G(v))$ is a Gelfand pair and apart from the two exceptional representations $\pi _v$ and $\pi _{v'}$ , every spherical representation of G admits a nonzero $\mathrm {Fix}_G(w)$ -invariant vector for all $w\in V(T)$ . In addition, the equivalence classes of spherical representations admitting a nonzero $\mathrm {Fix}_G(v)$ -invariant vector are in bijective correspondence with the interval $\lbrack -{1}/({d'-1});1\rbrack $ via the map $\phi _v:\pi \mapsto \varphi _\pi (\tau _v)$ where $\tau _v$ is an element of G such that $d(\tau _v v,v)=2$ and $d'$ is the degree of $v'$ . Under this correspondence, the exceptional spherical representation $\pi _v$ corresponds to $-{1}/({d'-1})$ and the trivial representation corresponds to $1$ . Finally, if $\pi $ is a nonexceptional spherical representation of G,

$$ \begin{align*} \phi_{v'}(\pi)=\frac{d(d'-1)}{d'(d-1)}\phi_v(\pi) + \frac{d-d'}{d'(d-1)}.\end{align*} $$

Remark 2.4. The author wishes to point out that certain extrema of the real interval corresponding to spherical functions of positive types found in the literature are incorrect when G has two orbits of vertices. The incorrect values provided by [Ama03, BK88] for instance are based on earlier inaccuracies [IP83] that were already pointed out in [CMoS94, Remark $2$ , page $243$ ]. The values provided in Theorem 2.3 can be computed from [Mat69].

To describe the special representations, let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 3$ , let $e\in E(T)$ and let $G\leq \operatorname {\mathrm {Aut}}(T)$ be an edge-transitive closed subgroup acting transitively on the boundary $\partial T$ . We define $\mathcal {L}(e)$ as the subspace of $\mathrm {Fix}_{G}(e)$ -right invariant square-integrable functions $\varphi : G \rightarrow \mathbb C$ satisfying

$$ \begin{align*} \int_{\mathrm{Fix}_G(v)}\varphi(gk)\,d \mu(k)=0\quad \forall g\in G \quad\forall v\in e. \end{align*} $$

Notice that $\mathcal {L}(e)$ is a closed left-invariant subspace of $L^2(G)$ and let $\sigma $ be the unitary representation of G defined by $\sigma (t)\varphi (g)=\varphi (t^{-1}g)$ for all $g,t\in G$ , for all $\varphi \in \mathcal {L}(e)$ . If G is transitive on the vertices of T, we choose an inversion $h\in G$ of the edge e and consider the linear map $\nu :\mathcal {L}(e) \rightarrow \mathcal {L}(e)$ defined by $\nu (\varphi )(g)=\varphi (gh)$ for all $\varphi \in \mathcal {L}(e)$ , for all $g\in G$ . This map is well defined since for all $\varphi \in \mathcal {L}(e)$ , for all $g\in G$ , for all $v\in e$ ,

$$ \begin{align*} \int_{\mathrm{Fix}_G(v)} (\nu\varphi)(gk)\,d \mu(k)&=\int_{\mathrm{Fix}_G(v)} \varphi(gkh)\,d \mu(k)\\ &=\int_{\mathrm{Fix}_G(v)} \varphi(ghh^{-1}kh)\,d \mu(k)\\ &=\int_{\mathrm{Fix}_G(h^{-1}v)} \varphi(ghk)\,d \mu(k)=0. \end{align*} $$

On the other hand, since every element of $\mathcal {L}(e)$ is $\mathrm {Fix}_G(e)$ -right invariant, note that $\nu $ is an involution and that it does not depend on our choice of inversion of the edge e. For each $\epsilon \in \{-1,1\}$ , let $\mathcal {L}(e)_{\epsilon }$ be the eigenspace of $\nu $ corresponding to the eigenvalue $\epsilon $ and let $\sigma ^\epsilon $ be the restriction of $\sigma $ to $\mathcal {L}(e)_{\epsilon }$ . We are now ready to state the classification of special representations.

Theorem 2.5 [FTN91, Ch. III, Section 2], [Mat77, Section 5.6].

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 3$ , let $e\in E(T)$ and let $G\leq \operatorname {\mathrm {Aut}}(T)$ be a closed noncompact subgroup acting transitively on the boundary $\partial T$ . Every special representation of G is square-integrable and admits a $\mathrm {Fix}_G(f)$ -invariant vector for every $f\in E(T)$ . Furthermore:

1. (1) If G acts transitively on $V(T)$ , $(\sigma ^{-1},\mathcal {L}(e)_{-1})$ and $(\sigma ^{+1},\mathcal {L}(e)_{+1})$ are representatives of the two equivalence classes of special representations.

2. (2) If G has two orbits on $V(T)$ , $(\sigma ,\mathcal {L}(e))$ is a representative of the unique equivalence class of special representations.

3 The Radu groups

Let T be a $(d_0,d_1)$ semi-regular tree with $d_0,d_1\geq 4$ and associated bipartition $V(T)=V_0\sqcup V_1$ , and let $\operatorname {\mathrm {Aut}}(T)^+$ denote the group of type-preserving automorphisms of T that is the set of automorphisms of T leaving $V_0$ and $V_1$ invariants. The purpose of this section is to recall the classification of Radu groups [Rad17]. For this purpose, we set

$$ \begin{align*} \mathcal{H}_T=\{ G\leq\operatorname{\mathrm{Aut}}(T)\mid G\mbox{ is closed and } 2\mbox{-transitive on }\partial T \} \end{align*} $$

and

$$ \begin{align*} \mathcal{H}_T^+=\{ G\leq\operatorname{\mathrm{Aut}}(T)^+\mid G\mbox{ is closed and } 2\mbox{-transitive on }\partial T \}. \end{align*} $$

If $d_0 \not = d_1$ , notice that every automorphism of T is type-preserving, so $\mathcal {H}_T^+= \mathcal {H}_T$ . We recall that for each vertex $v\in V(T)$ , the stabiliser $\mathrm {Fix}_G(v)$ of v acts on the set $E(v)$ of edges containing v, and that the image of $\mathrm {Fix}_G(v)$ in $\operatorname {\mathrm {Sym}}(E(v))$ for this projection map (which we denote by $\underline {G}(v)$ ) is called the local action of G at v. Furthermore, we recall in the light of [BM00, Lemma 3.1.1], that every group $G\in \mathcal {H}_T^+$ is transitive on $V_0$ and $V_1$ . Thus, all the groups $\underline {G}(v)$ with $v\in V_0$ (respectively $v\in V_1$ ) are permutation isomorphic to the same group $F_0\leq \operatorname {\mathrm {Sym}}(d_0)$ (respectively $F_1\leq \operatorname {\mathrm {Sym}}(d_1)$ ). A group $G\in \mathcal {H}_T$ such that $\underline {G}(v)\cong F_t\geq \operatorname {\mathrm {Alt}}(d_t)$ for each vertex $v\in V_t(T)$ and for $t\in \{0,1\}$ is called a Radu group. For each vertex $v\in V(T)$ and each positive integer $r\in \mathbb N$ let

$$ \begin{align*}S(v,r)=\{w\in V(T)\mid d(v,w)=r\}\end{align*} $$

be the set of vertices of T at distance r from v.

Definition 3.1. A legal colouring $i:V(T)\rightarrow \mathbb N$ of T is the concatenation of a pair of maps

$$ \begin{align*}i_0:V_0\rightarrow\{1,\ldots,d_1\}\quad\mbox{and}\quad i_1:V_1\rightarrow\{1,\ldots,d_0\}\end{align*} $$

such that are bijections for all $v\in V_1$ and $w\in V_0$ .

Given a legal colouring i of T and an automorphism $g\in \operatorname {\mathrm {Aut}}(T)$ , the local action of g at a vertex $v\in V(T)$ is defined as

Remark 3.2. If $d_0=d_1$ , the tree T is a regular tree and this notion of legal colouring and local action of an element is different from the notion of legal colouring and local action used to define the universal Burger–Mozes groups in [BM00]. Indeed, with our definition, the closed subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ of all automorphisms of trees $g\in G$ such that $\sigma _{(i)}(g,v)=\operatorname {\mathrm {id}} \text { for all } v\in V$ is not transitive on the set of vertices of T (not even transitive on $V_0$ ). It is good to note, however, that this colouring approach is compatible with that of Smith [Smi17].

Now, let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ and let i be a legal colouring of T. For each vertex $v\in V(T)$ and each finite set $Y\subseteq \mathbb N$ let

$$ \begin{align*}S_Y(v)=\bigcup_{r\in Y} S(v,r)\end{align*} $$

and for each set of vertices $B\subseteq V(T)$ let

$$ \begin{align*}\operatorname{\mathrm{Sgn}}_{(i)}(g,B)=\prod_{w\in B}^{}\operatorname{\mathrm{sgn}}(\sigma_{(i)}(g,w))\end{align*} $$

where $\operatorname {\mathrm {sgn}}(\sigma _{(i)}(g,w))$ is the sign of the local action $\sigma _{(i)}(g,w)$ of the automorphism g at w for the legal colouring i.

Definition 3.3. For all (possibly empty) finite sets $Y_0, Y_1$ of $\mathbb N$ and every legal colouring i of T, we set

$$ \begin{align*} G^+_{(i)}(Y_0,Y_1)=\left\{g\in \operatorname{\mathrm{Aut}}(T)^+\ \middle\vert\,\kern0.1pt \begin{array}{@{}l@{}} \operatorname{\mathrm{Sgn}}_{(i)}(g,S_{Y_0}(v))=1 \mbox{ for each }v\in V_{t_0}\\ \operatorname{\mathrm{Sgn}}_{(i)}(g,S_{Y_1}(v))=1 \mbox{ for each }v\in V_{t_1} \end{array}\right\}, \end{align*} $$

where $t_0=\max (Y_0)\ \pmod{\,2}$ , $t_1=(1+\max (Y_1))\ \pmod{\,2}$ and $\max (\varnothing )=0$ .

Remark 3.4. The choices of $t_0$ and $t_1$ are made in such a way that the vertices of $S_{Y_0}(v)$ with $v\in V_{t_0}$ at maximal distance from v and the vertices of $S_{Y_1}(w)$ with $w\in V_{t_1}$ at maximal distance from w have opposite types.

Note that $G_{(i)}^+(\varnothing ,\varnothing )=\operatorname {\mathrm {Aut}}(T)^+$ is the full group of type-preserving automorphisms and that $G^+_{(i)}(\{0\},\{0\})$ is a subgroup of each $G^+_{(i)}(Y_0,Y_1)$ . Furthermore, if T is a d-regular tree note that $G^+_{(i)}(\{0\},\{0\})$ is conjugate to $U(\operatorname {\mathrm {Alt}}(d))^+$ where $G^+=G\cap \operatorname {\mathrm {Aut}}(T)^+$ and $U(\operatorname {\mathrm {Alt}}(d))$ is the universal Burger–Mozes group of the alternating group see [BM00]. As we recall below, when $d_0,d_1\geq 6$ , every simple Radu group is of the form $G_{(i)}^+(Y_0,Y_1)$ for some finite $Y_0,Y_1\subseteq \mathbb N$ and some legal colouring i of T. Furthermore, every Radu group G belongs to a finite chain $H_n\geq \cdots \geq H_0$ with $n\in \{0,1,2,3\}$ such that $H_n=G$ , $\lbrack H_t: H_{t-1} \rbrack =2$ for all t and $H_0$ is conjugate in $\operatorname {\mathrm {Aut}}(T)^+$ to one of these $G_{(i)}^+(Y_0,Y_1)$ . In particular, Mackey’s machinery (see the Appendix) allows one to obtain a description of the irreducible representations of any Radu group from the irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ and vice versa. We now recall more precisely the results proved in [Rad17] that are used in this paper.

Theorem [Rad17, Theorem A].

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ and let i be a legal colouring of T. Then, for all finite subsets $Y_0, Y_1\subseteq \mathbb N$ the group $G^+_{(i)}(Y_0,Y_1)$ belongs to $\mathcal {H}^+_T$ and is abstractly simple.

To formulate the following results we let $G^{(\infty )}$ denote the intersection of all normal cocompact closed subgroups of the locally compact group G. Furthermore, we recall from [BM00, Proposition 3.1.2] that $H^{(\infty )}$ belongs to $\mathcal {H}^+_T$ and is topologically simple for all $H\in \mathcal {H}^+_T$ (in our cases, these groups are even abstractly simple). Finally, we let $\mathcal {G}_{T}^+(i)$ be the set of groups $G^+_{i}(Y_0,Y_1)$ with nonempty finite $Y_0, Y_1\subseteq \mathbb N$ such that $y=\max (Y_t)\ \pmod{\,2}$ for each $y\in Y_t$ with $y\geq \max (Y_{1-t})$ ( $t\in \{0,1\}$ ).

Theorem [Rad17, Theorem B].

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 6$ , let i be a legal colouring, and let $G\in \mathcal {H}^+_T$ be such that $\underline {G}(v)\cong F_0\geq \operatorname {\mathrm {Alt}}(d_0)$ and ${\underline {G}(w)\cong F_1\geq \operatorname {\mathrm {Alt}}(d_1)}$ for each $v\in V_0$ , $w\in V_1$ . Then, $G^{(\infty )}$ is conjugate in $\operatorname {\mathrm {Aut}}(T)^+$ to an element of $\mathcal {G}_{T}^+(i)$ and $\lbrack G: G^{(\infty )}\rbrack \in \{1,2,4\}$ .

When T is a d-regular tree, a similar result holds for all $G\in \mathcal {H}_T- \mathcal {H}_T^+$ .

Theorem [Rad17, Corollary C].

Let T be a d-regular tree with $d\geq 6$ and let i be a legal colouring and let $G\in \mathcal {H}_{T}- \mathcal {H}^+_T$ be such that $\underline {G}(v)\cong F\geq \operatorname {\mathrm {Alt}}(d)$ for each $v\in V(T)$ . Then, $G^{(\infty )}$ is conjugate to ${G}^+_{(i)}(Y,Y)$ for some finite subset Y of $\mathbb N$ and ${\lbrack G: G^{(\infty )} \rbrack \in \{2,4,8\}}$ .

Finally, the following theorem follows from Radu’s description of Radu groups.

Theorem 3.5. Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 6$ and let i be a legal colouring of T. Every Radu group G is contained in a finite chain $H_n\geq \cdots \geq H_0$ with $n\in \{0,1,2,3\}$ such that $H_n=G$ , $\lbrack H_t: H_{t-1} \rbrack =2$ for all t and $H_0$ is conjugate in $\operatorname {\mathrm {Aut}}(T)^+$ to some $G_{(i)}^+(Y_0,Y_1)$ .

4 Cuspidal representations of the simple Radu groups

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ and let $V(T)=V_0\sqcup V_1$ be the associated bipartition. Let i be a legal colouring of T and let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets. We recall from Section 3 that $G_{(i)}^+(Y_0,Y_1)$ (Definition 3.3) is a simple Radu group and that every simple Radu group is of this form when $d_0,d_1\geq 6$ . The purpose of this section is to describe the irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ and to show that this group is uniformly admissible, hence CCR.

We recall from Section 1 that the irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ split into three categories. These are either spherical, special or cuspidal. A classification of the spherical and special representations of any subgroup $G\leq \operatorname {\mathrm {Aut}}(T)$ acting $2$ -transitively on the boundary is already given in Section 2. In particular, this classification applies to $G_{(i)}^+(Y_0,Y_1)$ . Our current purpose is to give a description of the cuspidal representations of these groups. As announced in Section 1, our idea is to take advantage of the framework developed in [Sem23] and the description of these groups provided by Radu. The abstraction of Ol’shanskii’s framework expressed in [Sem23, Theorem A] provides a way to classify the representations at certain ‘depths’ where the depth is a generalisation of the ‘spherical/special/cuspidal’ framework, that is, larger depth means that more compact open subgroups have no nonzero invariant vector. This framework as well as the associated notion of depth depends on the existence and choice (assuming the existence) of a basis of neighbourhoods of the identity satisfying a certain factorisation property. Such a basis is given in [Sem23] for groups of automorphisms of trees satisfying the property $\mathrm {IP} _k$ as defined in [BEW15] and yielding to a classification of their cuspidal representations at sufficiently large depth depending on k. Since every Radu group G is known to satisfy the property $\mathrm {IP} _k$ for some sufficiently large k depending on G, this classification applies to the Radu groups. However, this approach only leads to a description of all the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ when $k=1$ , that is, when G satisfies the Tits independence property, because for that choice of basis of neighbourhoods of the identity, the set of the associated representations at depth $0$ contains some cuspidal representations (increasingly many as k grows). The purpose of this section is to go beyond this result by exhibiting the basis of neighbourhoods of the identity of the group $G_{(i)}^+(Y_0,Y_1)$ that satisfies the factorisation property at all positive depths, and whose associated representations at depth $0$ are exactly the spherical and special representations. This is done by relying on the local conditions given by the Definition 3.3 provided by Radu’s work rather than the property $\mathrm {IP}_k$ and leads to a complete description of the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ rather than a partial one; see Section 4.4.

4.1 Preliminaries

We start by recalling the axiomatic framework developed in [Sem23] (see that paper for details). Let G be a unimodular totally disconnected locally compact group, let $\mathcal {B}$ denote the set of compact open subgroups of G, $P(\mathcal {B})$ denote the power set of $\mathcal {B}$ and let

$$ \begin{align*}\mathcal{C}: \mathcal{B}\rightarrow P(\mathcal{B})\end{align*} $$

be the map sending a compact open subgroup to its conjugacy class in G. Let $\mathcal {S}$ be a basis of neighbourhoods of the identity consisting of compact open subgroups of G and let $\mathcal {F}_{\mathcal {S}}=\{ \mathcal {C}(U)\mid U\in \mathcal {S}\}$ . We equip $\mathcal {F}_{\mathcal {S}}$ with the partial order given by the reverse inclusion of representatives ( $\mathcal {C}(U)\leq \mathcal {C}(V)$ if there exists $\tilde {U}\in \mathcal {C}(U)$ and $\tilde {V}\in \mathcal {C}(V)$ such that $\tilde {V}\subseteq \tilde {U}$ ). For a poset $(P, \leq )$ and an element $x\in P$ , we recall that the height of x in $(P,\leq )$ is $L_x-1$ , where $L_x$ is the maximal length of a strictly increasing chain in $P_{\leq x}=\{y\in P\mid y\leq x\}$ if such a maximal length exists and we say that the height is infinite otherwise.

Definition 4.1. A basis of neighbourhoods of the identity $\mathcal {S}$ consisting of compact open subgroups of G is called a generic filtration of G if the height of every element in $\mathcal {F}_{\mathcal {S}}$ is finite.

Remark 4.2. Every unimodular totally disconnected locally compact group has a generic filtration, namely the compact open subgroups U with $\mu (U)\leq 1$ for some Haar measure $\mu $ .

Every generic filtration $\mathcal {S}$ of G splits as a disjoint union $\mathcal {S}=\bigsqcup _{l\in \mathbb N}\mathcal {S}\lbrack l \rbrack $ , where $\mathcal {S}\lbrack l\rbrack $ denotes the set of elements $U\in \mathcal {S}$ such that $\mathcal {C}(U)$ has height l in $\mathcal {F}_{\mathcal {S}}$ . The elements of $\mathcal {S}\lbrack l \rbrack $ are called the elements at depth l. Since $\mathcal {S}$ is a basis of neighbourhoods of the identity consisting of compact open subgroups of G, notice that for every irreducible representation $\pi $ of G, there exists a group $U\in \mathcal {S}$ such that $\pi $ admits a nonzero U-invariant vector. In particular, for every irreducible representation $\pi $ of G, there exists a smallest nonnegative integer $l_\pi \in \mathbb N$ such that $\pi $ admits nonzero U-invariant vectors for some $U\in \mathcal {S}\lbrack l_\pi \rbrack $ . This $l_\pi $ is called the depth of $\pi $ with respect to $\mathcal {S}$ . The key notion developed in [Sem23] is the notion of factorisation at depth l for a generic filtration $\mathcal {S}$ , which we now recall.

Definition 4.3. Let G be a nondiscrete unimodular totally disconnected locally compact group, let $\mathcal {S}$ be a generic filtration of G and let l be a strictly positive integer. We say that $\mathcal {S}$ factorises at depth l if the following conditions hold.

1. (1) For all $U\in \mathcal {S}\lbrack l \rbrack $ and every V in the conjugacy class of an element of $\mathcal {S}$ such that $V \not \subseteq U$ , there exists W in the conjugacy class of an element of $\mathcal {S}\lbrack l -1\rbrack $ such that

   $$ \begin{align*}U\subseteq W \subseteq V U=\{vu\mid u\in U, v\in V\}.\end{align*} $$

2. (2) For all $U\in \mathcal {S}\lbrack l \rbrack $ and every V in the conjugacy class of an element of $\mathcal {S}$ , the set

   $$ \begin{align*} N_G(U, V)= \{g\in G \mid g^{-1}Vg\subseteq U\} \end{align*} $$
   is compact.

Furthermore, the generic filtration $\mathcal {S}$ of G is said to factorise $^+$ at depth l if in addition for all $U\in \mathcal {S}\lbrack l\rbrack $ and every W in the conjugacy class of an element of $\mathcal {S}\lbrack l-1\rbrack $ such that $U\subseteq W$ ,

$$ \begin{align*} W\subseteq N_G(U,U) =\{g\in G \mid g^{-1}Ug\subseteq U\}. \end{align*} $$

Since G is unimodular, notice that the set $N_G(U,U)$ coincides with the normaliser $N_G(U)$ of U in G.

The relevance of this notion is given by [Sem23, Theorem A], which leads to a description of the irreducible representations at height l in terms of a family of irreducible representations of finite groups called $\mathcal {S}$ -standard representations (Definition 4.11 below), if the generic filtration factorises $^+$ at depth l.

4.2 Generic filtration for $G^+_{(i)}(Y_0,Y_1)$

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ and let $V(T)=V_0\sqcup V_1$ be the associated bipartition. Let i be a legal colouring of T and let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets. The purpose of this section is to provide a generic filtration for $G_{(i)}^+(Y_0,Y_1)$ . To do this, let $\mathfrak {T}_0$ be the family of subtrees of T defined by

$$ \begin{align*}\mathfrak{T}_{0}=\{B_T(v,r)\mid v\in V(T), r\geq 1\}\sqcup\{B_T(e,r)\mid e\in E(T), r\geq 0\}\end{align*} $$

and consider the basis of neighbourhoods of the identity given by the fixators of these trees

$$ \begin{align*} \mathcal{S}_{0}=\{\mathrm{Fix}_G(\mathcal{T})\mid \mathcal{T}\in \mathfrak{T}_{0}\}. \end{align*} $$

Definition 4.4. A group $G\leq \operatorname {\mathrm {Aut}}(T)$ is said to satisfy the hypothesis H ₀ if for all $\mathcal {T},\mathcal {T'}\in \mathfrak {T}_{0}$ ,

(H ₀) $$\begin{align} \mathrm{Fix}_G(\mathcal{T}')\leq \mathrm{Fix}_G(\mathcal{T})\quad\mbox{if and only if }\mathcal{T}\subseteq \mathcal{T}'. \end{align}$$

Lemma 4.5 [Sem23, Lemma 4.12].

Let $G\leq \operatorname {\mathrm {Aut}}(T)$ be a closed nondiscrete unimodular subgroup satisfying the hypothesis H₀ . Then, $\mathcal {S}_{0}$ is a generic filtration of G and the sets $\mathcal {S}_{0}\lbrack l \rbrack $ can be described as follows:

• • if l is even, $\mathcal {S}_{0}\lbrack l\rbrack = \{\mathrm {Fix}_G(B_T(e,\tfrac {l}{2}))\mid e\in E(T)\};$

• • if l is odd, $\mathcal {S}_{0}\lbrack l \rbrack =\{ \mathrm {Fix}_G(B_T(v,(({l+1})/{2})))\mid v\in V(T)\}.$

We come back to our case $G=G^+_{(i)}(Y_0,Y_1)$ .

Lemma 4.6. Let $G = G^+_{(i)}(Y_0,Y_1)$ and $\mathcal {T}$ be a complete finite subtree of T. Then, $\mathrm {Fix}_G(\mathcal {T})$ does not fix any vertices outside of $\mathcal {T}$ . In particular, $G^+_{(i)}(Y_0,Y_1)$ satisfies the hypothesis (H₀).

Proof. Since $H = G^+_{(i)}(\{0\},\{0\})$ is a subgroup of $G^+_{(i)}(Y_0,Y_1)$ , it suffices to prove that $\mathrm {Fix}_G(\mathcal {T})$ does not fix any vertices outside of $\mathcal {T}$ . If $\mathcal {T}$ has a single vertex, the result is trivial. Now, suppose that $\mathcal {T}$ contains at least one edge. For each leaf w of $\mathcal {T}$ , consider the unique vertex v of T that is adjacent to w and let

$$ \begin{align*}T(v,w)=\{x\in V(T): d_T(x,v) < d_T(x,w)\}.\end{align*} $$

By choice of w, one has that $\mathcal {T}\subseteq T(v,w)\cup \{w\}$ and since H satisfies Tits’ independence property, $\mathrm {Fix}_H(T(v,w))$ has a local action $\operatorname {\mathrm {Alt}}(d-1)$ at w, where w has degree d. In particular, since $d\geq 4$ , $\mathrm {Fix}_H(\mathcal {T})\subseteq \mathrm {Fix}_H(T(v,w)\cup \{w\})=\mathrm {Fix}_H(T(v,w))$ does not fix any neighbour of w other than v and thus does not fix any vertices outside of $T(v,w)\cup \{w\}$ . Since any vertex of T that does not belong to $\mathcal {T}$ must be outside of one of these $T(v,w)\cup \{w\}$ , it follows that $\mathrm {Fix}_H(\mathcal {T})$ does not fix any vertices outside of $\mathcal {T}$ and the result follows.

In particular, Lemma 4.5 ensures that $\mathcal {S}_0$ is a generic filtration of $G^+_{(i)}(Y_0,Y_1)$ .

4.3 Factorisation

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ and let $V(T)=V_0\sqcup V_1$ be the associated bipartition. Let i be a legal colouring of T and let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets. We have shown in Section 4.2 that $\mathcal {S}_0$ is a generic filtration of $G_{(i)}^+(Y_0,Y_1)$ . The purpose of the present section is to prove that this generic filtration factorises $^+$ at all depths $l\geq 1$ .

We start with some notation that is used in the proof (see Figure 1). For any two distinct vertices $v,w\in V(T)$ , let $\lbrack v,w\rbrack $ be the unique geodesic between v and w. Suppose that $d(v,w)=n$ , let $v=v_0,v_1,\ldots ,v_n=w$ be the sequence of vertices corresponding to $\lbrack v,w\rbrack $ in T, let

$$ \begin{align*} p_{\lbrack v,w\rbrack}: \lbrack v,w\rbrack -\{v\}\,\longrightarrow\, \lbrack v,w\rbrack: v_i\, \mapsto\, v_{i-1} \end{align*} $$

and let

$$ \begin{align*} T(v,w)&= \{x\in V(T)\mid d_T(x,p_{\lbrack v,w\rbrack }(w))<d_T(x,w)\}\\ &= \{x\in V(T)\mid d_T(x,v_{n-1})<d_T(x,w)\}. \end{align*} $$

Figure 1 The set $T(v,w)$ . Colour available online.

The following intermediate result is the key ingredient needed to prove the factorisation of the generic filtration $\mathcal {S}_0$ of $G_{(i)}^+(Y_0,Y_1)$ at all depths $l\geq 1$ .

Proposition 4.7. For all $l,l'\in \mathbb N$ such that $l\geq 1$ and $l'\geq l$ , for all U in the conjugacy class of an element of $\mathcal {S}_0\lbrack l \rbrack $ and every V in the conjugacy class of an element of $\mathcal {S}_0\lbrack l'\rbrack $ such that $V\not \subseteq U$ , there exists a subgroup $W\in \mathcal {S}_0\lbrack l-1\rbrack $ such that $U\subseteq W\subseteq VU$ .

Proof. To shorten the proof and to make the argument clearer, parts of the reasoning are proved in Lemmas 4.8 and 4.9 below. Since the proof is quite long and technical, we first give an idea of its structure. We begin the proof by identifying the group W from U and V. We then prove that every element of W decomposes as a product of an element of V and an element of U. The proof of this decomposition is the technical part. It is achieved by a compactness argument taking advantage from the fact that $G_{(i)}^+(Y_0,Y_1)$ is defined by local action conditions.

Let $G= G_{(i)}^+(Y_0,Y_1)$ . As announced at the beginning of the proof, we start by identifying W. Notice that $\mathfrak {T}_0$ is stable under the action of G. Furthermore, for every $g\in \operatorname {\mathrm {Aut}}(T)^+$ and for every subtree $\mathcal {T}$ of T, we have that $g\mathrm {Fix}_G(\mathcal {T})g^{-1}=\mathrm {Fix}_G(g\mathcal {T})$ . In particular, there exist $\mathcal {T}$ , $\mathcal {T}'\in \mathfrak {T}_0$ such that $U=\mathrm { Fix}_G(\mathcal {T})$ and $V=\mathrm {Fix}_G(\mathcal {T}')$ . Since $V\not \subseteq U$ , notice that $\mathcal {T}\not \subseteq \mathcal {T'}$ . If l is even, Lemma 4.5 ensures that $\mathcal {T}=B_T(e,\tfrac {l}{2})$ for some edge $e\in E(T)$ . Furthermore, since $\mathcal {T}\not \subseteq \mathcal {T'}$ and since $l'\geq l$ , there exists a unique vertex $v\in e$ such that $\mathcal {T}'\subseteq T(v,w)\cup B_T(v,\tfrac {l}{2})$ , where w denotes the other vertex of e. In this case, we let $\mathcal {T}_W=B_T(v,\tfrac {l}{2})$ . However, if l is odd, Lemma 4.5 ensures that $\mathcal {T}=B_T(w,({l+1})/{2})$ for some vertex $w\in V(T)$ . Furthermore, since $\mathcal {T}\not \subseteq \mathcal {T'}$ and since $l'\geq l$ , there exists a unique vertex $v\in B_T(w,1)-\{w\}$ such that $\mathcal {T}'\subseteq T(v,w)\cup B_T(\{v,w\},({l-1})/{2})$ . In that case, we let $\mathcal {T}_W=B_T(\{v,w\},({l-1})/{2})$ . In both cases, we set $W=\mathrm {Fix}_G( \mathcal {T}_W)$ . Notice by construction that $W\in \mathcal {S}_0\lbrack l-1\rbrack $ and that $U\subseteq W$ (since $\mathcal {T}_W\subseteq \mathcal {T}$ ). Our purpose is therefore to show that $W\subseteq V U$ . To this end, let $\alpha \in W$ and let us show the existence of an element $\alpha _0\in U$ such that . We start by explaining why the existence of $\alpha _0$ settles the proof. Indeed, if $\alpha _0$ exists, notice that the automorphism $\alpha _1=\alpha _0^{-1}\circ \alpha $ is an element of G for which . In particular, we have that $\alpha _1\in \mathrm {Fix}_G(\mathcal {T}')$ , $\alpha _0\in \mathrm {Fix}_G(\mathcal {T})$ and by construction $\alpha =\alpha _0 \circ \alpha _1$ , which proves that $W\subseteq UV$ . Applying the inverse map on both sides of the inclusion, we obtain that $W\subseteq VU$ which settles the proof.

Now, let us prove the existence of $\alpha _0$ . As announced at the beginning of the proof, we are going to use a compactness argument taking advantage of the fact that $G_{(i)}^+(Y_0,Y_1)$ is defined by local actions conditions. More precisely, we define a descending chain of nonempty compact sets $\Omega _n\subseteq \operatorname {\mathrm {Aut}}(T)^+$ and an increasing chain of finite subtrees $\mathcal {R}_n$ of T such that $T=\bigcup _{n\in \mathbb N} \mathcal {R}_n$ and such that for all $h\in \Omega _n$ :

• • $h\in \mathrm {Fix}_G(\mathcal {T})$ and ;

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h,S_{Y_0}(v))=1$ for all v in $V_{t_0}\cap \mathcal {R}_n$ ;

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h,S_{Y_1}(v))=1$ for all v in $V_{t_1}\cap \mathcal {R}_n$ .

We recall that in the above, $t_0=\max (Y_0)\,\mod 2$ , $t_1=(1+\max (Y_1))\ \pmod{\,2}$ and $\max (\varnothing )=0$ . Let us first show that this settles the existence of $\alpha _0$ . Since the $\Omega _n$ form a descending chain of nonempty compact sets in a Hausdorff space, we obtain $\bigcap _{n\in \mathbb N} \Omega _n\not =\varnothing $ . Let $\alpha _0 \in \bigcap _{n\in \mathbb N} \Omega _n$ . Since $\alpha _0\in \Omega _0$ , notice that , . To see that $\alpha _0$ is as desired, all that remains is to show that $\alpha _0\in G_{(i)}^+(Y_0,Y_1)$ . However, for every $v\in V_{t}(T)$ , there exists a positive integer $n\in \mathbb N$ such that $v\in \mathcal {R}_n$ and since $\alpha _0\in \Omega _{n}$ , we have that $\operatorname {\mathrm {Sgn}}_{(i)}(\alpha _0,S_{Y_t}(v))=1$ . This proves that $\alpha _0$ is as desired.

Figure 2 The tree $\mathcal {T}_W$ and the geodesic $\gamma $ . Colour available online.

What remains is to define the descending chain of nonempty compact sets $\Omega _n\subseteq \operatorname {\mathrm {Aut}}(T)^+$ . Suppose that $\max (Y_0)\leq \max (Y_1)$ (the proof for $\max (Y_1)\leq \max (Y_0)$ is similar). Let $\gamma $ be the smallest geodesic of T that contains both the centre of $\mathcal {T}$ and the centre of $\mathcal {T}'$ , and is oriented from $\mathcal {T}$ to $\mathcal {T}'$ (note that the centre is either a vertex or an edge depending on the values of l and $l'$ ). Since $\mathcal {T}\not \subseteq \mathcal {T}'$ and since $l'\geq l$ , notice that $\gamma $ contains at least two vertices (see Figure 2). The increasing chain of finite subtrees $\mathcal {R}_n$ of T such that $T=\bigcup _{n\in \mathbb N} \mathcal {R}_n$ that we are going to use is $\mathcal {R}_n=B_T(\gamma ,n)$ . We let

Since $\alpha \in \mathrm {Fix}_G(\mathcal {T}_W)$ and since $\mathcal {T}_W$ contains all vertices of $\mathcal {T}\cap \mathcal {T}'$ , notice that $\Omega _{-1}$ is not empty. Now, since $\max (Y_0)\leq \max (Y_1)$ , notice that there exists a unique $r\in \mathbb N$ such that $\max (Y_0)+2r\leq \max (Y_1)\leq \max (Y_0)+2r +1$ (where one of these inequalities is an equality). We let

$$ \begin{align*} \Omega_0=\left\{h\in \Omega_{-1}\ \middle\vert \begin{array}{@{\kern3pt}l@{}} \operatorname{\mathrm{Sgn}}_{(i)}(g,S_{Y_0}(v))=1 \mbox{ for each }v\in B_T(\gamma,2r)\cap V_{t_0}\\ \operatorname{\mathrm{Sgn}}_{(i)}(g,S_{Y_1}(v))=1 \mbox{ for each }v\in B_T(\gamma,0)\cap V_{t_1} \end{array}\right\}. \end{align*} $$

Lemma 4.8 below ensures that this set is not empty. From there, we define the sets $\Omega _n$ by induction on n. For every $n\geq 1$ , let $h_n$ be an element of $\Omega _{n-1}$ and let

For this induction to make sense, it is important for $\Omega _{n}$ to be not empty for all $n\geq 1$ . This is proved by Lemma 4.9 below, which ensures that $\Omega _{n}$ is a nonempty compact set. The result follows.

Our current purpose is to prove Lemmas 4.8 and 4.9. To this end, we introduce some formalism that is used in both proofs. For all $v\in V(T)$ , we need an automorphism $h_{(v)}\in \operatorname {\mathrm {Aut}}(T)^+$ that is used to create an element of $\omega _{n+1}$ from an element of $\Omega _n$ . We start by choosing four functions:

$$ \begin{align*} \phi_0 : \{1,\ldots,d_0\}\longrightarrow \operatorname{\mathrm{Sym}}(d_0): k \mapsto \phi_0(k); \end{align*} $$

$$ \begin{align*} \phi_1 : \{1,\ldots,d_1\}\longrightarrow \operatorname{\mathrm{Sym}}(d_1): k \mapsto \phi_1(k); \end{align*} $$

$$ \begin{align*} \tilde{\phi}_0 : \{1,\ldots,d_0\}\times\{1,\ldots,d_0\}\longrightarrow \operatorname{\mathrm{Sym}}(d_0): (k,l) \mapsto \tilde{\phi}_0(k,l); \end{align*} $$

$$ \begin{align*} \tilde{\phi}_1 : \{1,\ldots,d_1\}\times\{1,\ldots,d_1\}\longrightarrow \operatorname{\mathrm{Sym}}(d_1): (k,l) \mapsto \tilde{\phi}_1(k,l), \end{align*} $$

such that $\phi _t(k)$ is an odd permutation of $\operatorname {\mathrm {Sym}}(d_t)$ that fixes k and $\tilde {\phi }_t(k,l)$ is an odd permutation of $\operatorname {\mathrm {Sym}}(d_t)$ that fixes k and l.

If $v\in V(T)-\gamma $ , we choose $w\in \gamma $ and let $h_{(v)}\in \operatorname {\mathrm {Aut}}(T)^+$ be such that:

1. (1) $h_{(v)}\in \mathrm {Fix}_{\operatorname {\mathrm {Aut}}(T)^+}(T(p_{\lbrack w,v\rbrack }(v),v))$ ;

2. (2) $\sigma _{(i)}(h_{(v)},v)=\phi _t(i(p_{[w,v]}(v)))$ , where $t\in \{0,1\}$ is such that $v\in V_t$ .

Notice that for all $v\in V(T)-\gamma $ and every $w,w'\in \gamma $ , $p_{\lbrack w,v\rbrack }(v) = p_{\lbrack w',v\rbrack }(v)$ (recall the definition of $p_{\lbrack w,v\rbrack }$ from Section 4.3) so that our choice of $w\in \gamma $ does not change the two properties that $h_{(v)}$ must satisfy (see Figure 3).

Figure 3 The automorphism $h_{(v)}$ . Colour available online.

If $v\in \gamma $ , we have two cases. Remember that $\gamma $ has at least two vertices. If v is an end of $\gamma $ , let w be the unique vertex of $\gamma $ that is adjacent to v and choose an automorphism $h_{(v)}\in \operatorname {\mathrm {Aut}}(T)^+$ such that:

1. (1) $h_{(v)}\in \mathrm {Fix}_{\operatorname {\mathrm {Aut}}(T)^+}(T(w,v))$ ;

2. (2) $\sigma _{(i)}(h_{(v)},v)=\phi _t(i(w))$ , where $t\in \{0,1\}$ is such that $v\in V_t$ .

However, if v is not an end of $\gamma $ , let $w_1,w_2$ be the two neighbours of v that belong to $\gamma $ and choose an automorphism $h_{(v)}\in \operatorname {\mathrm {Aut}}(T)^+$ such that:

1. (1) $h_{(v)}\in \mathrm {Fix}_{\operatorname {\mathrm {Aut}}(T)^+}(T(w_1,v)\cup T(w_2,v))$ ;

2. (2) $\sigma _{(i)}(h_{(v)},v)=\tilde {\phi }_t(i(w_1),i(w_2))$ , where $t\in \{0,1\}$ is such that $v\in V_t$ .

We are now ready to prove Lemmas 4.8 and 4.9.

Lemma 4.8. The set

$$ \begin{align*} \Omega_0=\left\{h\in \Omega_{-1}\ \middle\vert\ \begin{array}{@{}l@{}} \operatorname{\mathrm{Sgn}}_{(i)}(g,S_{Y_0}(v))=1 \mbox{ for each }v\in B_T(\gamma,2r)\cap V_{t_0}\\ \operatorname{\mathrm{Sgn}}_{(i)}(g,S_{Y_1}(v))=1 \mbox{ for each }v\in B_T(\gamma,0)\cap V_{t_1} \end{array}\right\}. \end{align*} $$

is not empty.

Proof. We recall that $\max (Y_0)\leq \max (Y_1)$ , that $r\in \mathbb N$ is the unique integer such that $\max (Y_0)+2r\leq \max (Y_1)\leq \max (Y_0)+2r +1$ , that ${t_0=\max (Y_0) \ \pmod{\,2}}$ and that ${t_1=(\max (Y_1)+1) \ \pmod{\,2}}$ . Remember from the proof of Proposition 4.7 that $\Omega _{-1}$ is not empty and let $h_0\in \Omega _{-1}$ . We are going to modify the element $h_0$ with the automorphisms $h_{(v)}$ to get an element of $\Omega _{0}$ . A concrete example of the procedure is given on a $4$ -regular tree with $Y_0=\{0\}$ and $Y_1=\{1,2\}$ by Figures 4, 5 and 6. In these figures:

• • the hollow vertices are those considered by the current and previous steps;

• • the circled vertices are those for which we wish to change the sign $\operatorname {\mathrm {Sgn}}_{(i)}(g,S_{Y_0}(v))$ or $\operatorname {\mathrm {Sgn}}_{(i)}(g,S_{Y_1}(v))$ (depending on the step) without affecting the sign of other hollow vertices;

• • the vertices boxed in squares are the vertices for which a change of the local action is applied to achieve the desired change of sign (note that for our choice $Y_0=\{0\}$ , these vertices are also the circled vertices).

Figure 4 Step I of the proof of Lemma 4.8. Colour available online.

Figure 5 Step II of the proof of Lemma 4.8. Colour available online.

Figure 6 Step III of the proof of Lemma 4.8. Colour available online.

Let $\{w_{0,0},\ldots ,w_{0,m_0}\}$ be the set of vertices $w\in B_T(\gamma ,0)\cap V_{t_0}$ such that ${\operatorname {\mathrm {Sgn}}_{(i)}(h_0,S_{Y_0}(w))=-1}$ . For all $j=0,1,\ldots ,m_0$ , we choose a vertex

$$ \begin{align*}v_{0,j}\in \bigcap_{w\in \gamma-\{w_{0,j}\}}{}T(w_{0,j},w)\end{align*} $$

such that $d(v_{0,j},w_{0,j})=\max (Y_0)$ . In particular, notice that $v_{0,j}\in S_{Y_0}(w_{0,j})$ but that ${v_{0,j}\not \in S_{Y_0}(w)}$ for every $w \in B_T(\gamma ,0)\cap V_{t_0}-\{w_{0,j}\} $ . Furthermore, since $\operatorname {\mathrm {Sgn}}_{(i)}(h_0,S_{Y_0}(w_{0,j}))=-1$ , , , and due to the form of $\mathcal {T}$ and $\mathcal {T}'$ , the vertices $v_{0,j}$ must be such that the automorphisms $h_{(v_{0,0})},\ldots , h_{(v_{0,m_0})}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. Indeed, if $v_{0,j}$ is outside the trees $\mathcal {T}$ and $\mathcal {T}'$ or is a leaf of either of these trees, the conclusion is clear. Furthermore, $v_{0,j}$ can not be an interior vertex of $\mathcal {T}$ or $\mathcal {T}'$ , since $\operatorname {\mathrm {Sgn}}_{(i)}(h_0,S_{Y_0}(w_{0,j})) = -1$ and $h_0\in \Omega _{-1}$ . In particular, if $v_{0,j}$ was an interior vertex of $\mathcal {T}$ , $w_{0,j}$ would be too far inside $\mathcal {T}$ to have $\operatorname {\mathrm {Sgn}}_{(i)}(h_0,S_{Y_0}(w_{0,j})) = -1$ as $h_0$ fixes pointwise a large enough ball around $w_{0,j}$ and if $v_{0,j}$ was an interior vertex of $\mathcal {T}'$ , $w_{0,j}$ would be too far inside $\mathcal {T}'$ to have $\operatorname {\mathrm {Sgn}}_{(i)}(h_0,S_{Y_0}(w_{0,j})) = -1$ as $h_0$ agrees with $\alpha $ on a large enough ball around $w_{0,j}$ . This proves as desired that the automorphisms $h_{(v_{0,0})},\ldots , h_{(v_{0,m_0})}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. In particular, the automorphism

$$ \begin{align*}h_{0,0}= h_0 \circ h_{(v_{0,0})}\circ \cdots\circ h_{(v_{0,m_0})}\end{align*} $$

satisfies , and

$$ \begin{align*}\operatorname{\mathrm{Sgn}}_{(i)}(h_{0,0},S_{Y_0}(w))=1\quad \forall w\in \gamma\cap V_{t_0}.\end{align*} $$

If $r\not =0$ , we iterate this procedure. For every $1\leq \nu \leq 2r$ , let $\{w_{\nu ,0},\ldots ,w_{\nu ,m_{\nu }}\}$ be the set of vertices $w\in B_T(\gamma ,\nu )\cap V_{t_0}$ such that

$$ \begin{align*}\operatorname{\mathrm{Sgn}}_{(i)}(h_{\nu-1,0},S_{Y_0}(w))=-1.\end{align*} $$

For all $j=0,1,\ldots ,m_\nu $ , we choose a vertex

$$ \begin{align*}v_{\nu,j}\in \bigcap_{w\in B_T(\gamma,\nu)\cap V_{t_0}-\{w_{\nu,j}\}}{}T(w_{\nu,j},w)\end{align*} $$

such that $d(v_{\nu ,j},w_{\nu ,j})=\max (Y_0)$ . Hence, notice that $v_{\nu ,j}\in S_{Y_0}(w_{\nu ,j})$ but that $v_{\nu ,j}\not \in S_{Y_0}(w)$ for every $w\in B_T(\gamma ,\nu )\cap V_{t_0}-\{w_{\nu ,j}\}$ . Furthermore, notice that the automorphisms $h_{(v_{\nu ,0})}, \ldots , h_{(v_{\nu ,m_\nu })}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. In particular, the automorphism

$$ \begin{align*}h_{\nu,0}=h_{\nu-1,0} \circ h_{(v_{\nu,0})}\circ \cdots\circ h_{(v_{\nu,m_\nu})}\end{align*} $$

satisfies that , and

$$ \begin{align*}\operatorname{\mathrm{Sgn}}_{(i)}(h_{\nu,0},S_{Y_0}(w))=1\quad \forall w\in B_T(\gamma,\nu)\cap V_{t_0}.\end{align*} $$

Consider the element $h_{2r,0}$ that we have just constructed. This element behaves as desired for the condition given by $Y_0$ on the vertices of $B_T(\gamma ,2r)\cap V_{t_0}$ . However, nothing ensures that the condition given by $Y_1$ on the vertices of $B_T(\gamma ,0)\cap V_{t_1}$ is yet satisfied. We now take care of this task. Let $\{w_0,\ldots ,w_m\}$ be the set of vertices $w\in B_T( \gamma ,0)\cap V_{t_1}$ such that $\operatorname {\mathrm {Sgn}}_{(i)}(h_{2r,0},S_{Y_1}(w))=-1$ . For all $j=0,1,\ldots ,m$ , let ${v_j\in \bigcap _{w\in \gamma -\{w_j\}} T(w_j,w)}$ such that $d(v_j,w_j)=\max (Y_1)$ . In particular, notice that $v_{j}\in S_{Y_1}(w_{j})$ but that $v_{j}\not \in S_{Y_1}(w)$ for every $w\in B_T(\gamma ,0) \cap V_{t_1}-\{w_j\}$ . Furthermore, since $\max (Y_0)+2r\leq \max (Y_1)$ , notice from Remark 3.4 that $v_j\not \in S_{Y_0}(v)$ for every $v\in B_T(\gamma ,2r)\cap V_{t_0}$ . However, just as before, the automorphisms $h_{(v_{0})}, \ldots , h_{(v_{m})}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. In particular, $h_{1}= h_{2r,0}\circ h_{(v_{0})}\circ \cdots \circ h_{(v_{m})}$ satisfies:

• • and ;

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h_{1},S_{Y_0}(w))=1 \text { for all } w\in B_T(\gamma ,2r)\cap V_{t_0};$

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h_{1},S_{Y_1}(w))=1 \text { for all } w\in B_T(\gamma ,0)\cap V_{t_1}.$

This proves that $h_1\in \Omega _0$ and therefore that $\Omega _0$ is not empty.

Lemma 4.9. For all $n\geq 1$ , the set

is a nonempty compact subset of $\operatorname {\mathrm {Aut}}(T)^+$ .

Figure 7 Step I of the proof of Lemma 4.9. Colour available online.

Figure 8 Step II of the proof of Lemma 4.9. Colour available online.

Proof. We show that $\Omega _n$ is not empty by induction. Lemma 4.8 ensures that $\Omega _0$ is not empty. Suppose that $\Omega _{n-1}$ is not empty and let $h_n\in \Omega _{n-1}$ be the automorphism appearing in the definition of $\Omega _n$ . Just as in the proof of Lemma 4.8, we are going to modify $h_n$ with the automorphisms $h_{(v)}$ to obtain an element of $\Omega _{n}$ . A concrete example of the procedure is given by Figures 7 and 8 on a $4$ -regular tree with $Y_0=\{0\}$ and even $\max {(Y_1)}$ (with the same conventions as before and where the vertices concerned by the current step are highlighted in dotted areas). Let $\{\tilde {w}_0,\ldots ,\tilde {w}_{k}\}$ be the set of vertices $w$ that belong to $ B_T(\gamma ,n+2r)\cap V_{t_0}$ and such that $\operatorname {\mathrm {Sgn}}_{(i)}(h_n,S_{Y_0}(w))=-1$ . For each $j=0,1,\ldots ,k$ , we choose a vertex ${\tilde {v}_{j}\in \bigcap _{w\in B_T(\gamma ,n+2r)\cap V_{t_0}-\{\tilde {w}_j\}}{}T(\tilde {w}_{j},w)}$ such that $d(\tilde {v}_{j},\tilde {w}_{j})=\max (Y_0)$ . Hence, notice that $\tilde {v}_j\not \in S_{Y_0}(w)$ for all ${w\in B_T(\gamma ,n+2r)\cap V_{t_0}-\{\tilde {w}_j\}}$ . Furthermore, since $\max (Y_1)-1\leq 2r+\max (Y_0)$ , notice from Remark 3.4 that $\tilde {v}_j\not \in S_{Y_1}(w)$ for all ${w\in B_T(\gamma ,n-1)\cap V_{t_1}}$ . Furthermore, since ${\operatorname {\mathrm {Sgn}}_{(i)}(h_{n},S_{Y_0}(\tilde {w}_{j}))=-1}$ , , , and due to the form’s of $\mathcal {T}$ and $\mathcal {T}'$ , notice that the automorphisms $h_{(\tilde {v}_{0})}, \ldots , h_{(\tilde {v}_{k})}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. Indeed, if $\tilde {v}_{j}$ is outside the trees $\mathcal {T}$ and $\mathcal {T}'$ or is a leaf of either of these trees, the conclusion is clear. However, $\tilde {v}_{j}$ cannot be an interior vertex of $\mathcal {T}$ or $\mathcal {T}'$ , since $\operatorname {\mathrm {Sgn}}_{(i)}(h_n,S_{Y_0}(\tilde {w}_{j})) = -1$ and $h_n\in \Omega _{n-1}$ . In particular, if $\tilde {v}_{j}$ was an interior vertex of $\mathcal {T}$ , $\tilde {w}_{j}$ would be too far inside $\mathcal {T}$ for $\operatorname {\mathrm {Sgn}}_{(i)}(h_n,S_{Y_0}(\tilde {w}_{j})) = -1$ as $h_n$ fixes pointwise a large enough ball around $\tilde {w}_{j}$ and if $\tilde {v}_{j}$ was an interior vertex of $\mathcal {T}'$ , $\tilde {w}_{j}$ would be too far inside $\mathcal {T}'$ for $\operatorname {\mathrm {Sgn}}_{(i)}(h_n,S_{Y_0}(\tilde {w}_{j})) = -1$ as $h_n$ agrees with $\alpha $ on a large enough ball around $\tilde {w}_{j}$ . This proves as desired that the automorphisms $h_{(\tilde {v}_{0})}, \ldots , h_{(\tilde {v}_{k})}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. In particular, $\tilde {h}_n=h_{n} \circ h_{(\tilde {v}_{0})}\circ \cdots \circ h_{(\tilde {v}_{k})}$ satisfies:

• •

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h_{n},S_{Y_0}(w))=1 \text { for all } w\in B_T(\gamma ,n+2r)\cap V_{t_0};$

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h_{n},S_{Y_1}(w))=1 \text { for all } w\in B_T(\gamma ,n-1)\cap V_{t_1}.$

Now, let $\{w_0,\ldots ,w_m\}$ be the set of vertices $w\in B_T(\gamma ,n)\cap V_{t_1}$ such that $\operatorname {\mathrm {Sgn}}_{(i)}(\tilde {h}_n,S_{Y_1}(w))=-1$ . For each $j=0,1,\ldots ,m$ , choose $v_{j}\in \bigcap _{w\in B_T(\gamma ,n)-\{w_j\}} T(w_{j},w)$ such that $d(v_{j},w_{j})=\max (Y_1)$ . Since $\max (Y_0)+2r\leq \max (Y_1)$ , notice from Remark 3.4 that $v_j\not \in S_{Y_0}(w)$ for every $w\in B_T(\gamma ,n+2r)\cap V_{t_0}$ and $v_j\not \in S_{Y_1}(w)$ for every $w\in B_T(\gamma ,n)\cap V_{t_1}-\{w_j\}$ . Just as before, notice that the automorphisms $h_{(v_{0})}, \ldots ,h_{(v_{m})}$ fix $\mathcal {T}\cup \mathcal {T}'$ pointwise. In particular, $h_{n+1}=\tilde {h}_{n} \circ h_{(v_{0})}\circ \cdots \circ h_{(v_{m})}$ satisfies:

• •

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h_{n+1},S_{Y_0}(w))=1 \text { for all } w\in B(\gamma ,n+2r)\cap V_{t_0};$

• • $\operatorname {\mathrm {Sgn}}_{(i)}(h_{n+1},S_{Y_1}(w))=1 \text { for all } w\in B(\gamma ,n)\cap V_{t_1}.$

This proves that $\Omega _n$ is not empty. We now show that $\Omega _n$ is compact for every integer $n\geq 1$ . To this end, notice that $\Omega _n$ is a closed subset of $\operatorname {\mathrm {Aut}}(T)^+$ and that

$$ \begin{align*} \Omega_n \subseteq h_{n} \mathrm{Fix}_{\operatorname{\mathrm{Aut}}(T)^+}(B_T(\gamma,n-1+\max(Y_1))). \end{align*} $$

Since the right-hand side is a compact subset of $\operatorname {\mathrm {Aut}}(T)^+$ , the results follow.

Finally, we prove the result announced at the beginning of Section 4.3.

Theorem 4.10. The generic filtration $\mathcal {S}_{0}$ of $G^+_{(i)}(Y_0,Y_1)$ factorises $^+$ at every depth $l\geq 1$ .

Proof. Let $G=G^+_{(i)}(Y_0,Y_1)$ . To prove that $\mathcal {S}_0$ factorises $^+$ at depth $l\geq 1$ , we successively check the three conditions of Definition 4.3.

First, we must prove that for every U in the conjugacy class of an element of $\mathcal {S}_0\lbrack l \rbrack $ and every subgroup V in the conjugacy class of an element of $\mathcal {S}_0$ with $V \not \subseteq U$ , there exists a W in the conjugacy class of an element of $\mathcal {S}_0\lbrack l -1\rbrack $ such that

$$ \begin{align*}U\subseteq W \subseteq V U.\end{align*} $$

Let U and V be as above. By the definition of $\mathcal {S}_0$ and since $\mathfrak {T}_0$ is stable under the action of G, there exist two subtrees $\mathcal {T},\mathcal {T}'\in \mathfrak {T}_0$ such that $U=\mathrm {Fix}_G(\mathcal {T})$ and $V=\mathrm {Fix}_G(\mathcal {T}')$ . In particular, $\mathcal {T}'= B_T(o,r)$ , where o is either a vertex or an edge of T and r is a nonnegative integer. Let v be the vertex of $\mathcal {T}$ at maximal distance from o and notice since $V \not \subseteq U$ that $d_T(v,o)> r$ . Now let $\gamma $ be an infinite geodesic ray starting at v and containing o, let $\mathcal {P}$ be the tree centred at $\gamma (d_T(v,o) + l)$ of radius $r+l$ and let $V'=\mathrm {Fix}_G(\mathcal {P})$ . By construction, one has that $\mathcal {T}'\subseteq \mathcal {P}$ so that $V'\subseteq V$ . Furthermore, $V'$ has depth greater than l and since $v\not \in \mathcal {P}$ , Lemma 4.6 ensures that $V'\not \subseteq U$ . Now, Proposition 4.7 ensures the existence of a subgroup $W\in \mathcal {S}_0\lbrack l-1\rbrack $ such that

$$ \begin{align*}U\subseteq W \subseteq V'U\subseteq VU.\end{align*} $$

This proves the first condition.

Next, we must prove that $N_{G}(U, V)= \{g\in {G} \mid g^{-1}Vg\subseteq U\}$ is compact for every V in the conjugacy class of an element of $\mathcal {S}_0$ . Just as before, notice that there exists a $\mathcal {T}'\in \mathfrak {T}_0$ such that $V = \mathrm {Fix}_G(\mathcal {T}')$ . Since G satisfies the hypothesis (H₀), notice that

$$ \begin{align*} N_{G}(U, V)&= \{g\in G \mid g^{-1}Vg\subseteq U\}=\{g\in G \mid g^{-1}\mathrm{Fix}_G(\mathcal{T}')g\subseteq \mathrm{Fix}_G(\mathcal{T})\}\\ &=\{g\in G \mid \mathrm{Fix}_G(g^{-1}\mathcal{T}')\subseteq \mathrm{Fix}_G(\mathcal{T})\}= \{g\in G \mid g\mathcal{T}\subseteq \mathcal{T}'\}. \end{align*} $$

In particular, since both $\mathcal {T}$ and $\mathcal {T}'$ are finite subtrees of T, $N_G(U,V)$ is a compact subset of G which proves the second condition.

Finally, we must prove for every W in the conjugacy class of an element of $\mathcal {S}_0\lbrack l-1\rbrack $ with $U\subseteq W$ that

$$ \begin{align*} W\subseteq N_G(U,U) =\{g\in G \mid g^{-1}Ug\subseteq U\}. \end{align*} $$

For the same reasons as before, there exists $\mathcal {R}\in \mathfrak {T}_0$ such that $W= \mathrm {Fix}_G(\mathcal {R})$ . Furthermore, since $U\subseteq W$ and since G satisfies the hypothesis (H₀), notice that $\mathcal {R}\subseteq \mathcal {T}$ . Moreover, since $\mathrm {Fix}_G(\mathcal {R})$ has depth $l-1$ , notice that $\mathcal {R}$ contains every interior vertex of $\mathcal {T}$ . Since G is unimodular and satisfies hypothesis (H₀), this implies that

$$ \begin{align*} \mathrm{Fix}_G(\mathcal{R})&\subseteq \{h\in G \mid h\mathcal{T}\subseteq \mathcal{T}\}= \{h\in G \mid \mathrm{Fix}_G(\mathcal{T})\subseteq\mathrm{Fix}_G(h\mathcal{T})\}\\ &=\{h\in G \mid h^{-1}\mathrm{Fix}_G(\mathcal{T})h\subseteq\mathrm{Fix}_G(\mathcal{T})\}= N_G(U,U), \end{align*} $$

which proves the third condition.

4.4 Description of cuspidal representations

The purpose of this section is to give a description of the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ . This is done by Theorem 4.12 below but requires some preliminaries. We refer to [Sem23] for proofs and details of the formalism.

Let G be a nondiscrete unimodular totally disconnected locally compact group G and let $\mathcal {S}$ be a generic filtration of G factorising $^+$ at depth l. Then, for every irreducible representation $\pi $ of G at depth l, [Sem23, Theorem A] ensures the existence of a unique conjugacy class $C_\pi \in \mathcal {F}_{\mathcal {S}}=\{\mathcal {C}(U)\mid U\in \mathcal {S}\}$ at height l such that $\pi $ admits nonzero U-invariant vectors for any $U\in C_\pi $ . The conjugacy class $C_\pi $ is called the seed of $\pi $ . We define the group of automorphisms $\operatorname {\mathrm {Aut}}_{G}(C)$ of the seed C as the quotient $N_G(U)/U$ corresponding to any $U\in C$ . This group $\operatorname {\mathrm {Aut}}_{G}(C)$ is finite and does not depend on our choice of $U\in C$ up to isomorphism. Now, let $p_{U}: N_G(U)\mapsto N_G(U)/U$ denote the quotient map, let

$$ \begin{align*} \tilde{\mathfrak{H}}_{\mathcal{S}}(U)= \{W \mid \text{there exists } g\in G\mbox{ such that }gWg^{-1}\in \mathcal{S}\lbrack l-1 \rbrack \mbox{ and } U \subseteq W \} \end{align*} $$

and set

$$ \begin{align*}\mathfrak{H}_{\mathcal{S}}(C)=\{ p_U(W)\mid W\in \tilde{\mathfrak{H}}_{\mathcal{S}}(U)\}.\end{align*} $$

Notice that $\mathfrak {H}_{\mathcal {S}}(C)$ does not depend on our choice of representative $U\in C$ .

Definition 4.11. An irreducible representation $\omega $ of $\operatorname {\mathrm {Aut}}_{G}(C)$ is called $\mathcal {S}$ -standard if it has no nonzero H-invariant vector for any $H\in \mathfrak {H}_{\mathcal {S}}(C)$ .

The importance of this notion is given by [Sem23, Theorem A], which ensures that the irreducible representations of G at depth l with seed C are obtained from the $\mathcal {S}$ -standard representations of $\operatorname {\mathrm {Aut}}_G(C)$ when $\mathcal {S}$ factorises $^+$ at depth l. More precisely, we recall that every irreducible representation $\omega $ of $\operatorname {\mathrm {Aut}}_G(C)\cong N_G(U)/U$ can be lifted to an irreducible representation $\omega \circ p_U$ of $N_G(U)$ with U acting trivially and with representation space $\mathcal {H}_{\omega }$ . The lifted representation can then be induced to G. The resulting representation

$$ \begin{align*}T(U,\omega)=\operatorname{\mathrm{Ind}}_{N_G(U)}^G(\omega\circ p_{U})\end{align*} $$

is an irreducible representation of G with seed $\mathcal {C}(U)$ . Conversely, if $\pi $ is an irreducible representation of G with seed C, notice that $\mathcal {H}_{\pi }^{U}$ is a nonzero $N_G(U)$ -invariant subspace of $\mathcal {H}_{\pi }$ for every $U\in C$ . In particular, the restriction is a representation of $N_G(U)$ whose restriction to U is trivial. This representation passes to the quotient group $N_G(U)/U$ and defines an $\mathcal {S}$ -standard representation $\omega _\pi $ of $\operatorname {\mathrm {Aut}}_{G}(C)$ .

We now come back to the case in which we are interested. Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ , let $V(T)=V_0\sqcup V_1$ be the associated bipartition and let

$$ \begin{align*}\mathfrak{T}_{0}=\{B_T(v,r)\mid v\in V(T), r\geq 1\}\sqcup\{B_T(e,r)\mid e\in E(T), r\geq 0\}.\end{align*} $$

Let i be a legal colouring of T, let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets and consider the group $G_{(i)}^+(Y_0,Y_1)$ . Let $G=G_{(i)}^+(Y_0,Y_1)$ . We have shown in Sections 4.2 and 4.3 that

$$ \begin{align*} \mathcal{S}_{0}=\{\mathrm{Fix}_G(\mathcal{T})\mid \mathcal{T}\in \mathfrak{T}_{0}\} \end{align*} $$

is a generic filtration of $G_{(i)}^+(Y_0,Y_1)$ that factorises $^+$ at all depths $l\geq 1$ . In particular, [Sem23, Theorem A] provides a bijective correspondence between irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ at depth l with seed $C \in \mathcal {F}_{\mathcal {S}_0}$ and the $\mathcal {S}_0$ -standard representations of $\operatorname {\mathrm {Aut}}_G(C)$ . We start by identifying these seeds and show that the cuspidal representations of G are precisely the irreducible representations of G at depth $l\geq 1$ with respect to $\mathcal {S}_0$ . In the light of Lemma 4.5, we consider the partition $\mathfrak {T}_0=\bigsqcup _{l\in \mathbb N}\mathfrak {T}_0\lbrack l\rbrack $ where:

• • $\mathfrak {T}_0\lbrack l\rbrack = \{B_T(e,\tfrac {l}{2})\mid e\in E(T)\}$ if l is even;

• • $\mathfrak {T}_0\lbrack l \rbrack =\{ B_T(v,(({l+1})/{2}))\mid v\in V(T)\}$ if l is odd.

Notice that for all $l\in \mathbb N$ , $\mathfrak {T}_0\lbrack l\rbrack $ is stable under the action of G. Furthermore, notice that if l is even, or if l is odd and G is transitive on the vertices, the set $\mathfrak {T}_0\lbrack l\rbrack $ consists of a single G-orbit. However, if l is odd and G has two orbits of vertices, notice that the set $\mathfrak {T}_0\lbrack l\rbrack $ consists of two G-orbits, namely $\{B_T(v,(({l+1})/{2}))\mid v\in V_0\}$ and $\{B_T(v,(({l+1})/{2}))\mid v\in V_1\}$ . In particular, in light of Lemma 4.6, there are either one or two elements of $\mathcal {F}_{\mathcal {S}_0}=\{\mathcal {C}(U)\mid U\in \mathcal {S}_0\}$ at height l and each such element is of the form

$$ \begin{align*}C=\{\mathrm{Fix}_G(\mathcal{T})\mid \mathcal{T}\in \mathcal{O}\},\end{align*} $$

where $\mathcal {O}$ is a G-orbit of $\mathfrak {T}_0\lbrack l \rbrack $ . We deduce easily that the irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ at depth $l\geq 1$ with respect to $\mathcal {S}_0$ are the cuspidal representations of G. Indeed, $\pi $ is an irreducible representation at depth $l\geq 1$ with respect to $\mathcal {S}_0$ if and only if $\pi $ does not admit a nonzero V-invariant vector for any V in a conjugacy class C at depth $0$ , that is, for any $V\in \{\mathrm {Fix}_G(e)\mid e\in E(T)\}$ . Now, let $\pi $ be a cuspidal representation of G, let $C_\pi \in \mathcal {F}_{\mathcal {S}_0}$ be the seed of $\pi $ , let $U\in C_\pi $ and let $\mathcal {T}\in \mathfrak {T}_0$ be such that $U=\mathrm {Fix}_G(\mathcal {T})$ . Since $\mathcal {S}_0$ factorises $^+$ at all depths $l\geq 1$ , [Sem23, Theorem A] ensures that $\pi $ is induced from an irreducible representation of $N_G(U)$ that passes to the quotient ${\operatorname {\mathrm {Aut}}_G(C_\pi )\cong N_G(U)/U}$ . Furthermore, since G satisfies hypothesis (H₀), notice that

$$ \begin{align*} N_G(U)= \{g\in G\mid gUg^{-1}=U\} &=\{g\in G\mid g\mathrm{Fix}_G(\mathcal{T})g^{-1}=\mathrm{Fix}_G(\mathcal{T})\}\\ &=\{g\in G\mid \mathrm{Fix}_G(g\mathcal{T})=\mathrm{Fix}_G(\mathcal{T})\}\\ &=\{g\in G\mid g\mathcal{T}=\mathcal{T}\}=\operatorname{\mathrm{Stab}}_G(\mathcal{T}) \end{align*} $$

is exactly the stabiliser of $\mathcal {T}$ in $G_{(i)}^+(Y_0,Y_1)$ . In particular, $\operatorname {\mathrm {Aut}}_G(C_\pi )$ can be identified with the automorphism group of $\mathcal {T}$ obtained by restricting the action of $\operatorname {\mathrm {Stab}}_G(\mathcal {T})$ to $\mathcal {T}$ . Moreover, since G satisfies hypothesis (H₀), notice that

$$ \begin{align*} \tilde{\mathfrak{H}}_{\mathcal{S}_0}(U)=\{\mathrm{Fix}_G(\mathcal{R})\mid \mathcal{R}\in \mathfrak{T}_0, \mathcal{R}\subsetneq \mathcal{T} \mbox{ and }\mathcal{R}\mbox{ is maximal for this property}\}, \end{align*} $$

and

$$ \begin{align*}\mathfrak{H}_{\mathcal{S}_0}(C_\pi)=\{ p_U(W)\mid W\in \tilde{\mathfrak{H}}_{\mathcal{S}_0}(U)\}\end{align*} $$

is the set of fixators (in $\operatorname {\mathrm {Aut}}_G(C_\pi )$ ) of subtrees $\mathcal {R}\in \mathcal {T}_0$ satisfying $\mathcal {R}\subsetneq \mathcal {T}$ and that are maximal for this property. In particular, the $\mathcal {S}_0$ -standard representations of $\operatorname {\mathrm {Aut}}_G(C_\pi )$ are the irreducible representations of the group of automorphisms of $\mathcal {T}$ obtained by restricting the action of $\operatorname {\mathrm {Stab}}_G(\mathcal {T})$ and that do not admit any nonzero invariant vector for the fixator of any subtree $\mathcal {R}$ of $\mathcal {T}$ that belongs to $\mathfrak {T}_0$ and is maximal for this property. The above discussion together with [Sem23, Theorem A] leads to the following description of the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ .

Theorem 4.12. Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ , let i be a legal colouring of T, let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets, let $G=G_{(i)}^+(Y_0,Y_1)$ , consider the generic filtration $\mathcal {S}_0$ of G (defined in Section 4.2) and let us use the above notation. Then, the cuspidal representations of G are exactly the irreducible representations at depth $l\geq 1$ with respect to $\mathcal {S}_{0}$ . Furthermore, if $\pi $ is a cuspidal representation at depth l, we have the following.

• • $\pi $ has no nonzero $\mathrm {Fix}_G(\mathcal {R})$ -invariant vector for any $\mathcal {R}\in \bigsqcup _{r<l}\mathfrak {T}_0\lbrack r\rbrack $ .

• • There exists a unique conjugacy class $C_\pi \in \mathcal {F}_{\mathcal {S}_0}$ at height l such that $\pi $ admits a nonzero U-invariant vector for any (hence for all) $U\in C_\pi $ . Equivalently, there exists a unique G-orbit $\mathcal {O}$ of $\mathfrak {T}_0\lbrack l \rbrack $ such that $\pi $ admits a nonzero $\mathrm {Fix}_G(\mathcal {T})$ -invariant vector for any (hence for all) $\mathcal {T}\in \mathcal {O}$ . Furthermore, $\mathcal {O}$ is the only orbit of $\mathfrak {T}_0$ under the action of G such that $C_\pi =\{\mathrm {Fix}_G(\mathcal {T})\mid \mathcal {T}\in \mathcal {O}\}$ .

• • If $\mathcal {O}$ is the unique G-orbit of $\mathfrak {T}_0\lbrack l \rbrack $ corresponding to $\pi $ and if $\mathcal {T}\in \mathcal {O}$ , $\pi $ admits a nonzero diagonal matrix coefficient supported in $\operatorname {\mathrm {Stab}}_G(\mathcal {T})$ . In particular, $\pi $ is square-integrable and its equivalence class is isolated in the unitary dual $\widehat {G}$ for the Fell topology.

Furthermore for each $C\in \mathcal {F}_{\mathcal {S}_0}$ at height $l\geq 1$ with corresponding G-orbit $\mathcal {O}$ in $\mathfrak {T}_0\lbrack l \rbrack $ , that is, $C=\{\mathrm {Fix}_G(\mathcal {T})\mid \mathcal {T}\in \mathcal {O}\}$ , there exists a bijective correspondence between the equivalence classes of irreducible representations of G with seed C and the equivalence classes of $\mathcal {S}_0$ -standard representations of $\operatorname {\mathrm {Aut}}_G(C)$ . More precisely, for every $\mathcal {T}\in \mathcal {O}$ , the following hold.

1. (1) If $\pi $ is a cuspidal representation of G with seed C, $(\omega _\pi , \mathcal {H}_{\pi }^{\mathrm {Fix}_G(\mathcal {T})})$ is an $\mathcal {S}_0$ -standard representation of $\operatorname {\mathrm {Aut}}_{G}(C)$ such that

   $$ \begin{align*} \pi \cong T(\mathrm{Fix}_G(\mathcal{T}),\omega_\pi)=\operatorname{\mathrm{Ind}}_{\operatorname{\mathrm{Stab}}_G(\mathcal{T})}^{G}(\omega_\pi \circ p_{\mathrm{Fix}_G(\mathcal{T})}). \end{align*} $$

2. (2) If $\omega $ is an $\mathcal {S}_0$ -standard representation of $\operatorname {\mathrm {Aut}}_{G}(C)$ , the representation $T(\mathrm {Fix}_G(\mathcal {T}),\omega )$ is a cuspidal representation of G with seed in C.

Furthermore, if $\omega _1$ and $\omega _2$ are $\mathcal {S}_0$ -standard representations of $\operatorname {\mathrm {Aut}}_{G}(C)$ , we have that $T(\mathrm {Fix}_G(\mathcal {T}),\omega _1)\cong T(\mathrm {Fix}_G(\mathcal {T}),\omega _2)$ if and only if $\omega _1\cong \omega _2$ . In particular, the above two constructions are inverse of one another.

4.5 Existence of cuspidal representations

As in the previous sections, let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ , let $V(T)=V_0\sqcup V_1$ be the associated bipartition and let

$$ \begin{align*}\mathfrak{T}_{0}=\{B_T(v,r)\mid v\in V(T), r\geq 1\}\sqcup\{B_T(e,r)\mid e\in E(T), r\geq 0\}.\end{align*} $$

Let i be a legal colouring of T, let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets and consider the group $G_{(i)}^+(Y_0,Y_1)$ . Write $G=G_{(i)}^+(Y_0,Y_1)$ if this leads to no confusion. We have shown in Sections 4.2 and 4.3 that

$$ \begin{align*} \mathcal{S}_{0}=\{\mathrm{Fix}_G(\mathcal{T})\mid \mathcal{T}\in \mathfrak{T}_{0}\} \end{align*} $$

is a generic filtration of $G_{(i)}^+(Y_0,Y_1)$ factorising $^+$ at all depths $l\geq 1$ . In particular, [Sem23, Theorem A] provides a bijective correspondence between the irreducible representations of $G_{(i)}^+(Y_0,Y_1)$ at depth l with seed $C \in \mathcal {F}_{\mathcal {S}_0}$ and the $\mathcal {S}_0$ -standard representations of $\operatorname {\mathrm {Aut}}_G(C)$ . This leads to a description of the cuspidal representations of $G_{(i)}^+(Y_0,Y_1)$ ; see Theorem 4.12. However, none of these results yet ensures the existence of a cuspidal representation of $G_{(i)}^+(Y_0,Y_1)$ . The purpose of this section is to prove the existence of a cuspidal representation with seed C for each conjugacy class $C\in \mathcal {F}_{\mathcal {S}_0}$ at height $l\geq 1$ . From the description of cuspidal representations of $G^+_{(i)}(Y_0,Y_1)$ provided by Theorem 4.12, it is equivalent to prove the following theorem.

Theorem 4.13. Let $G=G^+_{(i)}(Y_0,Y_1)$ and let $C\in \mathcal {F}_{\mathcal {S}_0}$ be a conjugacy class at height $l\geq 1$ . Then, there exists an ${\mathcal {S}_0}$ -standard representation of $\operatorname {\mathrm {Aut}}_G(C)$ .

The proof of this theorem is gathered in the following results. We start by recalling a result from [Sem23].

Proposition 4.14 [Sem23, Proposition $2.29$ ].

Let T be a locally finite tree, let ${G\leq \operatorname {\mathrm {Aut}}(T)}$ be a closed subgroup, let $\mathcal {T}$ be a finite subtree of T and let $\{\mathcal {T}_1,\mathcal {T}_2,\ldots ,\mathcal {T}_s\}$ be a set of distinct finite subtrees of T contained in $\mathcal {T}$ such that $\mathcal {T}_i\cup \mathcal {T}_j=\mathcal {T}$ for every $i\not =j$ . Suppose that $\operatorname {\mathrm {Stab}}_G(\mathcal {T})$ acts by permutation on the set $\{\mathcal {T}_1,\mathcal {T}_2,\ldots ,\mathcal {T}_s\}$ and that $\mathrm {Fix}_G(\mathcal {T})\subsetneq \mathrm {Fix}_G(\mathcal {T}_i)\subsetneq \operatorname {\mathrm {Stab}}_G(\mathcal {T})$ . Then, there exists an irreducible representation of $\operatorname {\mathrm {Stab}}_G(\mathcal {T})/\mathrm {Fix}_G(\mathcal {T})$ without a nonzero $\mathrm {Fix}_G(\mathcal {T}_i)/\mathrm {Fix}_G(\mathcal {T})$ -invariant vector for every ${i=1,\ldots ,s}$ .

The following proposition ensures the existence of ${\mathcal {S}_0}$ -standard representations of $\operatorname {\mathrm {Aut}}_{G}(C)$ for every $C\in \mathcal {F}_{\mathcal {S}_0}$ with height $l\geq 2$ .

Proposition 4.15. Let $G=G^+_{(i)}(Y_0,Y_1)$ and let $C\in \mathcal {F}_{\mathcal {S}_0}$ be a conjugacy class at height $l\geq 2$ . Then, there exists an ${\mathcal {S}_0}$ -standard representation of $\operatorname {\mathrm {Aut}}_{G}(C)$ .

Proof. Since $l\geq 2$ , Lemma 4.5 ensures the existence of a complete finite subtree $\mathcal {T}$ of T containing a ball of radius $1$ around an edge such that $C=\mathcal {C}(\mathrm { Fix}_{G}(B_T(e,r)))$ . Furthermore, as observed in Section 4.4, we have that $\operatorname {\mathrm {Aut}}_G(C)\cong \operatorname {\mathrm {Stab}}_G(\mathcal {T})/\mathrm {Fix}_G(\mathcal {T})$ and

$$ \begin{align*} \mathfrak{H}_{\mathcal{S}_0}(\mathrm{Fix}_G(\mathcal{T}))&=\{\mathrm{Fix}_G(\mathcal{R})/\mathrm{Fix}_G(\mathcal{T})\mid \mathcal{R}\in \mathfrak{T}_0, \mathcal{R}\subsetneq \mathcal{T} \\ &\qquad\,\mbox{and } \mathcal{R}\mbox{ is maximal for this property}\}. \end{align*} $$

Now, let $\{\mathcal {T}_1,\ldots , \mathcal {T}_s\}$ be the set of maximal complete proper subtrees of $\mathcal {T}$ . These can all be obtained from $\mathcal {T}$ by removing the leaves adjacent to v, where v is a vertex of $\mathcal {T}$ that is not a leaf but such that all but one of its neighbours are leaves of $\mathcal {T}$ . Clearly, $\mathcal {T}_i\cup \mathcal {T}_j=\mathcal {T}$ for every $i\not =j$ and $\operatorname {\mathrm {Stab}}_G(\mathcal {T})$ acts by permutation on the set $\{\mathcal {T}_1,\mathcal {T}_2,\ldots ,\mathcal {T}_s\}$ . Furthermore, for each $i\in \{ 1,\ldots ,s\}$ , notice from Lemma 4.6 that

$$ \begin{align*}\mathrm{ Fix}_G(\mathcal{T})\subsetneq \mathrm{Fix}_G(\mathcal{T}_i).\end{align*} $$

Furthermore, since $\mathcal {T}$ belongs to $\mathfrak {T}_0$ and contains at least a ball of radius $1$ around an edge, notice that $\mathrm {Fix}_G(\mathcal {T}_i)\subseteq \operatorname {\mathrm {Stab}}_G(\mathcal {T})$ . However, this inclusion must be strict since $\operatorname {\mathrm {Stab}}_G(\mathcal {T})$ contains either the fixator of a vertex of $\mathcal {T}_i$ or the fixator of one of its edges. Now, Proposition 4.14 ensures the existence of an irreducible representation of $\operatorname {\mathrm {Stab}}_G(\mathcal {T})/\mathrm {Fix}_G(\mathcal {T})$ without a nonzero $\mathrm {Fix}_G(\mathcal {T}_i)/\mathrm {Fix}_G(\mathcal {T})$ -invariant vector for every $i=1,\ldots ,s$ . Since every proper subtree $\mathcal {R}\in \mathfrak {T}_0$ of $\mathcal {T}$ is contained in one of these $\mathcal {T}_i$ , this representation is an $\mathcal {S}_0$ -standard representation of $\operatorname {\mathrm {Stab}}_G(\mathcal {T})/\mathrm {Fix}_G(\mathcal {T})$ .

The next lemma treats the remaining case $l=1$ where Proposition 4.14 does not apply.

Lemma 4.16. Let $G=G^+_{(i)}(Y_0,Y_1)$ and let $C\in \mathcal {F}_{\mathcal {S}_0}$ be a conjugacy class at height $1$ . Then, there exists an $\mathcal {S}_{0}$ -standard representation of $\operatorname {\mathrm {Aut}}_{G}(C)$ .

Proof. Lemma 4.5 ensures the existence of a vertex $v\in V(T)$ such that ${C=\mathcal {C}(\mathrm {Fix}_{G}(B_T(v,1)))}$ . Let $\mathcal {T}=B_T(v,r)$ . We recall as observed in Section 4.4 that $\operatorname {\mathrm {Aut}}_G(C)\cong \operatorname {\mathrm {Stab}}_G(\mathcal {T})/\mathrm {Fix}_G(\mathcal {T})$ , where

$$ \begin{align*} \operatorname {\mathrm {Stab}}_G(\mathcal {T})=\{g\in G \mid g\mathcal {T}\subseteq \mathcal {T}\}= \{ g\in G \mid gv=v\}= \mathrm { Fix}_G(v).\end{align*} $$

In particular, $\operatorname {\mathrm {Aut}}_G(C)$ can be realised as a the group of automorphisms of $B_T(v,1)$ obtained by restricting the action of $\mathrm {Fix}_G(v)$ . Furthermore,

$$ \begin{align*} \mathfrak{H}_{\mathcal{S}_0}(\mathrm{Fix}_G(\mathcal{T}))&=\{\mathrm{Fix}_G(\mathcal{R})/\mathrm{Fix}_G(\mathcal{T})\mid \mathcal{R}\in \mathfrak{T}_0, \mathcal{R}\subsetneq \mathcal{T} \\ &\qquad\,\mbox{and } \mathcal{R}\mbox{ is maximal for this property}\}\\ &=\{\mathrm{Fix}_G(f)/\mathrm{Fix}_G(B_T(v,1))\mid f\in E(B_T(v,1))\}. \end{align*} $$

Let d be the degree of v in T, let $X=E(B_T(v,1))$ and let $e\in X$ . Since $\operatorname {\mathrm {Alt}}(d)\leq \underline {G}(v)$ and since $d\geq 4$ , notice that $\underline {G}(v)$ is $2$ -transitive. In particular, $\operatorname {\mathrm {Aut}}_{G}(C)$ is $2$ -transitive on X and [Sem23, Lemma 2.27] ensures the existence of an irreducible representation $\sigma $ of $\operatorname {\mathrm {Aut}}_{G}(C)$ without a nonzero $\mathrm {Fix}_{\operatorname {\mathrm {Aut}}_{G}(C)}(e)$ -invariant vector. Since $\mathrm {Fix}_G(v)$ is transitive on $E(B_T(v,1))$ , this representation does not admit a nonzero $\mathrm {Fix}_{\operatorname {\mathrm {Aut}}_{G}(C)}(f)$ -invariant vector for any $f\in E(B_T(v,1))$ . The lemma follows from the fact that $\mathrm { Fix}_{\operatorname {\mathrm {Aut}}_{G}(C)}(f)=\mathrm {Fix}_G(f)/\mathrm {Fix}_G(B_T(v,1))$ .

5 Simple Radu groups are CCR

Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 4$ . Let i be a legal colouring of T and let $Y_0,Y_1\subseteq \mathbb N$ be two finite subsets. The purpose of this section is to exploit the classification of the irreducible representations of $G^+_{(i)}(Y_0,Y_1)$ obtained from Sections 2 and 4.4 to prove that $G^+_{(i)}(Y_0,Y_1)$ is uniformly admissible and hence CCR.

We recall that a totally disconnected locally compact group G is uniformly admissible if for every compact open subgroup K, there exists a positive integer $k_K$ such that $\dim (\mathcal {H}_{\pi }^K)<k_K$ for every irreducible representation $\pi $ of G. In particular, uniformly admissible groups are CCR. The following classical result ensures that the spherical representations of any Radu group are uniformly admissible.

Theorem 5.1. Let T be a $(d_0,d_1)$ -semi-regular tree with $d_0,d_1\geq 3$ , let $G\leq \operatorname {\mathrm {Aut}}(T)$ be a closed noncompact subgroup acting transitively on the boundary of T and let $v\in V(T)$ . Then, for every integer $n\geq 1$ , there exists a constant $k_n\in \mathbb N$ such that $\mathrm {dim}(\mathcal {H}_{\pi }^{K_n})<k_n$ for every spherical representation $\pi $ of G admitting a nonzero $\mathrm {Fix}_G(v)$ -invariant vector and where $K_n=\mathrm { Fix}_G(B_T(v,n))$ .

Proof. Let $K=\mathrm {Fix}_G(v)$ and let $\mu $ be the Haar measure of G renormalised in such a way that $\mu (K)=1$ . Theorems 2.2 and 2.3 ensure that $(G,K)$ is a Gelfand pair and we observe that $\mathrm {dim}(\mathcal {H}_{\pi }^K)= 1.$ Now let $\xi $ be a unit vector of $\mathcal {H}_{\pi }^K$ and let $\eta $ be a unit vector of $\mathcal {H}_{\pi }^{K_n}.$ Notice that

$$ \begin{align*}\varphi_{\xi, \eta}: G\,\longrightarrow\, \mathbb C: g\, \mapsto\, \langle \pi(g)\xi,\eta\rangle\end{align*} $$

is a K-right invariant and $K_n$ -left invariant continuous function. However, since $\mathrm {dim}(\mathcal {H}_{\pi }^K)= 1$ , notice for all $g,h\in G$ that

$$ \begin{align*} \int_K\varphi_{\xi,\eta}(gkh)\,\mathrm{d}\mu(k)&=\bigg\langle \int_K \pi(gkh)\xi,\eta\bigg\rangle\\ &=\bigg\langle \pi(h)\xi,\int_K\pi(k^{-1})\pi(g^{-1})\eta\bigg\rangle\\ &=\langle \pi(h)\xi,\alpha(\eta, g)\xi\rangle=\overline{\alpha(\eta, g)}\varphi_{\xi,\xi}(h) \end{align*} $$

for some $\alpha (\eta , g)\in \mathbb C$ . However, $\varphi _{\xi ,\xi }(1_G)=1$ and hence $\overline {\alpha (\eta , g)}=\varphi _{\xi ,\eta }(g)$ . This implies for all $g,h\in G$ that

(5-1) $$ \begin{align} \int_K\varphi_{\xi,\eta}(gkh)\, d \mu(k)=\varphi_{\xi,\eta}(g)\varphi_{\xi,\xi }(h). \end{align} $$

Since $\varphi _{\xi ,\eta }$ is K-right invariant and $K_n$ -left invariant, notice that it can be realised as a function $\phi :Gv\rightarrow \mathbb C$ on the orbit $Gv$ of v in $V(T)$ that is constant on the $K_n$ -orbits of v. However, since $K_n$ is an open subgroup of the compact group K, the index of $K_n$ in K is finite. Since K is transitive on the boundary of the tree, this implies that $K_n$ has finitely many orbits on $\partial T$ . In particular, there exists an integer $N_n\geq \max \{2,n\}$ such that $\partial T(w,v)$ is contained in a single $K_n$ -orbit for all $w\in \partial B_T(v,N_n)$ , where $\partial T(w,v)$ is the set of ends of $T(w,v)=\{u\in V(T)\mid d_T(u,w)<d_T(u,v)\}$ that are not vertices. Now notice that the value of $\phi $ on every vertex $w\in Gv$ can be computed iteratively from the values that $\phi $ takes on vertices of $B_T(v,N_n)$ . More precisely, let w be any vertex in $Gv - B_T(v,N_n)$ and let u be the vertex of $Gv$ at distance $2$ from w that is closer to $B_T(v,N_n)$ than any other vertex of $Gv$ at distance $2$ from w. Let $t\in G$ be such that $d_T(v,t v)=2$ , $g\in G$ be such that $gv=u$ and note from Equation (5-1) (taking $h = t$ ) that

(5-2) $$ \begin{align} \int_K \phi(gkt v)\,d \mu (k) = \phi(u) \varphi_{\xi,\xi}(t). \end{align} $$

Now note that every vertex in $\{gkt v : k \in K\}$ is either in the same $K_n$ -orbit as w or is strictly closer to $B_T(v,N_n)$ than w. In particular, Equation (5-2) expresses $\phi (w)$ as a linear combination of the values that $\phi $ takes on vertices that are strictly closer to $B_T(v,N_n)$ than w. Now, for each of these vertices that do not already belong to $B_T(v,N_n)$ , we can apply the same reasoning and obtain the value that $\phi $ takes on it in terms of the values that $\phi $ takes on vertices that are even closer to $B_T(v,N_n)$ . Since $N_n$ has been chosen to be at least $2$ , this process must end at some point and one obtains an expression for $\phi (w)$ as a linear combination of the values that $\phi $ takes on the vertices of $B_T(v,N_n)$ . This implies that the space $\mathcal {L}_n$ of functions $\varphi :G\rightarrow \mathbb C$ that are K-right invariant, $K_n$ -left invariant and satisfy Equation (5-1) has finite dimension bounded by the cardinality $k_n$ of $B_T(v,N_n)$ . Furthermore, since $\pi $ is irreducible, notice that $\xi $ is cyclic and therefore that the linear map $\Psi _n: \mathcal {H}_{\pi }^{K_n}\rightarrow \mathcal {L}_n: \eta \rightarrow \overline {\varphi _{\xi ,\eta }}$ is injective. This proves as desired that $\dim (\mathcal {H}_{\pi }^{K_n})\leq \dim (\mathcal {L}_n)\leq k_n<+\infty $ .

In addition, the following classical result provides a bound for every special and cuspidal representation.

Theorem 5.2 [HC70, Corollary of Theorem 2].

Let G be a locally compact group, $\pi $ be an irreducible square-integrable representation of G and $K\leq G$ be a compact open subgroup. Then, there exists a positive integer $k_{K,\pi }$ depending on K and $\pi $ such that $\dim (\mathcal {H}_{\pi }^K)\leq k_{K,\pi }$ .

Putting it all together, we can prove the result announced at the beginning of the section.

Theorem 5.3. $G^+_{(i)}(Y_0,Y_1)$ is uniformly admissible, and hence CCR.

Proof. Let $G=G^+_{(i)}(Y_0,Y_1)$ . Let $K\leq G$ be a compact open subgroup, and let $v\in V$ . For each $n\in \mathbb N$ , let $K_n=\mathrm {Fix}_G(B_T(v,n))$ and notice that Theorem 5.1 ensures the existence of a positive integer $k_{n}$ such that $\dim (\mathcal {H}_{\pi }^{K_n})\leq k_{n}$ for every spherical representation $\pi $ of G. Now, for every $n\in \mathbb N$ , we let $\Sigma _n$ be the subset of all equivalence classes of nonspherical irreducible representations of G admitting nonzero $K_n$ -invariant vectors. The classification of special and cuspidal representations of G provided by Theorems 2.5 and 4.12 ensures that $\Sigma _n$ is finite. Furthermore, for every $\sigma \in \Sigma _n$ , Theorem 5.2 provides a constant $k_{\sigma , n}$ such that $\dim (\mathcal {H}_{\pi }^{K_n})\leq k_{\sigma , n}$ . In particular, for each $n\in \mathbb N$ , the constant $k^{\prime }_n=\max _{\sigma \in \Sigma _n} k_{\sigma , n}$ is finite and $\dim (\mathcal {H}_{\pi }^{K_n})< k^{\prime }_n$ for each special and cuspidal representation $\pi $ of G. It follows that $\dim (\mathcal {H}_{\pi }^{K_n})\leq \max \{k_{n}, k^{\prime }_{n}\}$ for all $\pi \in \widehat {G}$ and every $n\in \mathbb N$ . However, since $(K_n)_{n\in \mathbb N}$ is a basis of neighbourhoods of the identity, there exists some $N\in \mathbb N$ such that $K_N\subseteq K$ . It follows that

$$ \begin{align*}\dim(\mathcal{H}_{\pi}^K)\leq \max\{k_{N}, k^{\prime}_{N}\}\quad \forall \pi \in \widehat{G}.\\[-42pt]\end{align*} $$