A note on the rational homological dimension of lattices in positive characteristic

Abstract

We show via $\ell^2$ -homology that the rational homological dimension of a lattice in a product of simple simply connected Chevalley groups over global function fields is equal to the rational cohomological dimension and to the dimension of the associated Bruhat–Tits building.

1. Introduction

Let k be the function field of an irreducible projective smooth curve C defined over a finite field $\mathbb{F}_q$ . Let S be a finite non-empty set of (closed) points of C. Let ${\mathcal{O}}_S$ be the ring of rational functions whose poles lie in S. For each $p\in S,$ there is a discrete valuation $\nu_x$ of k such that $\nu_p(f)$ is the order of vanishing of f at p. The valuation ring ${\mathcal{O}}_p$ is the ring of functions that do not have a pole at p, that is

\begin{equation*}{\mathcal{O}}_S=\bigcap_{p\not\in S}{\mathcal{O}}_p.\end{equation*}

Let $\bar{k}$ denote the algebraic closure of k. Let $\mathbf{G}$ be an affine group scheme defined over $\bar{k}$ such that $\mathbf{G}(\bar{k})$ is almost simple. For each $p\in S,$ there is a completion $k_p$ of k and the group $\mathbf{G}(k_p)$ acts on the Bruhat–Tits building $X_p$ . Thus, we may embed $\mathbf{G}({\mathcal{O}}_S)$ diagonally into the product $\prod_{p\in S}\mathbf{G}(k_p)$ as an arithmetic lattice.

The rational cohomological dimension of a group $\Gamma$ is defined to be

\begin{equation*}\textrm{cd}_\mathbb{Q}(\Gamma)\,{:\!=}\, \sup\{n\colon H^n(\Gamma;M)\neq0,\ M\textrm{ a }\mathbb{Q}\Gamma\textrm{-module} \},\end{equation*}

the rational homological dimension is defined completely analogously as

\begin{equation*}\textrm{hd}_\mathbb{Q}(\Gamma)\,{:\!=}\, \sup\{n\colon H_n(\Gamma;M)\neq0,\ M\textrm{ a }\mathbb{Q}\Gamma\textrm{-module} \}.\end{equation*}

In [5], it is shown that $\textrm{cd}_\mathbb{Q}(\mathbf{G}({\mathcal{O}}_S))=\prod_{p\in S}\dim (X_p)$ . In light of this, Ian Leary asked the author what is $\textrm{hd}_\mathbb{Q}(\mathbf{G}({\mathcal{O}}_S))$ ?

Theorem A. Let $\mathbf{G}$ be a simple simply connected Chevalley group. Let k and ${\mathcal{O}}_S$ be as above. Then

\begin{equation*}\textrm{hd}_\mathbb{Q}(\mathbf{G}({\mathcal{O}}_S))=\textrm{cd}_\mathbb{Q}(\mathbf{G}({\mathcal{O}}_S))=\prod_{p\in S}\dim (X_p).\end{equation*}

More generally, we obtain the following.

Corollary B. Let $\Gamma$ be a lattice in a product of simple simply connected Chevalley groups over global function fields with associated Bruhat–Tits building X. Then $\textrm{hd}_\mathbb{Q}(\Gamma)=\textrm{cd}_\mathbb{Q}(\Gamma)=\dim (X)$ .

The author expects these results are well-known; however, they do not appear in the literature so we take the opportunity to record them here.

2. $\ell^2$ –homology and measure equivalence

Let $\Gamma$ be a group. Both $\Gamma$ and the complex group algebra $\mathbb{C} \Gamma$ act by left multiplication on the Hilbert space $\ell^2\Gamma$ of square-summable sequences. The group von Neumann algebra ${\mathcal{N}} \Gamma$ is the ring of $\Gamma$ -equivariant bounded operators on $\ell^2G$ . The non-zero divisors of ${\mathcal{N}} G$ form an Ore set and the Ore localization of ${\mathcal{N}} \Gamma$ can be identified with the ring of affiliated operators ${\mathcal{U}} \Gamma$ .

There are inclusions $\mathbb{Q}\Gamma\subseteq {\mathcal{N}} \Gamma\subseteq\ell^2\Gamma\subseteq{\mathcal{U}} \Gamma,$ and it is also known that ${\mathcal{U}} \Gamma$ is a self-injective ring which is flat over ${\mathcal{N}} \Gamma$ . For more details concerning these constructions, we refer the reader to [12] and especially to Theorem 8.22 of Section 8.2.3 therein. The von Neumann dimension and the basic properties we need can be found in [12, Section 8.3].

The $\ell^2$ -Betti numbers of a group $\Gamma$ , denoted $b_i^{(2)}(\Gamma)$ , are then defined to be the von-Neumann dimensions of the homology groups $H_i(\Gamma;\ {\mathcal{U}}\Gamma)$ . The following lemma is a triviality.

Lemma 2.1. Let $\Gamma$ be a discrete group and suppose that $b^{(2)}_i(\Gamma)>0$ . Then the homology group $H_i(\Gamma;\ {\mathcal{U}}\Gamma)$ is non-trivial.

Two countable groups $\Gamma$ and $\Lambda$ are said to be measure equivalent if there exist commuting, measure-preserving, free actions of $\Gamma$ and $\Lambda$ on some infinite Lebesgue measure space $(\Omega,m)$ , such that the action of each of the groups $\Gamma$ and $\Lambda$ admits a finite measure fundamental domain. The key examples of measure equivalent groups are lattices in the same locally compact group [6]. The relevance of this for us is the following deep theorem of Gaboriau.

Theorem 2.2. (Gaboriau’s Theorem [4]) Suppose a discrete group $\Gamma$ is measure equivalent to a discrete group $\Lambda$ . Then $b_p(\Gamma)=0$ if and only if $b_p(\Lambda)=0$ .

3. Proofs

Proof of Theorem A. We first note that the group $\Gamma:=\mathbf{G}({\mathcal{O}}_S)$ is measure equivalent to the product $\Lambda:=\prod_{p\in S}\mathbf{G}(\mathbb{F}_q[t_p])$ for some suitably chosen $t_p\in {\mathcal{O}}_p$ . By [13, Theorem 1.6] (see also [2,3,1]), the group $\mathbf{G}(\mathbb{F}_q[t_p])$ has one non-vanishing $\ell^2$ -Betti number in dimension $\dim(X_p)$ . Hence, by the Künneth formula $\Lambda$ has one non-vanishing $\ell^2$ -Betti number in dimension $d=\prod_{p\in S}\dim (X_p).$ Thus, by Gaboriau’s Theorem, the group $\Gamma$ has exactly one non-vanishing $\ell^2$ -Betti number in dimension d. It follows from Lemma 2.1 that $\textrm{hd}_\mathbb{Q}(\Gamma)\geq d$ . The reverse inequality follows from the fact that $\Gamma$ acts properly on the d-dimensional space $\prod_{p\in S}\dim (X_p)$ .

Proof of Corollary B. The proof of the corollary is entirely analogous. First, we split $\mathbf{G}$ into a product of simple groups $\prod_{i=1}^n\mathbf{G}_i$ corresponding to the decomposition of the Bruhat–Tits building $X=\prod_{i=1}^nX_i$ . Let $\Lambda_i$ be a lattice in $\mathbf{G}_i$ and let $\Lambda=\prod_{i=1}^n\Lambda_i$ . Each $\Lambda_i$ has a non-vanishing $\ell^2$ -Betti number in dimension $\dim (X_i)$ . In particular, $\Lambda$ has a non-vanishing $\ell^2$ -Betti number in dimension $\dim (X)=\prod_{i=1}^n\dim (X_i)$ . By Gaboriau’s Theorem, $\Gamma$ also has non-vanishing $\ell^2$ -Betti number in dimension $\dim (X)$ . It follows from Lemma 2.1 that $\textrm{hd}_\mathbb{Q}(\Gamma)\geq d$ . The reverse inequality follows from the fact that $\Gamma$ acts properly on the d-dimensional space $\prod_{p\in S} X_p$ .

Remark 3.1. A similar argument can be applied to lattices in products of simple simply connected algebraic groups over locally compact p-adic fields. One obtains the analogous result for such a lattice $\Gamma$ that $\textrm{cd}_\mathbb{Q} (\Gamma)=\textrm{hd}_\mathbb{Q} (\Gamma)=\dim (X)$ , where X is the associated Bruhat–Tits building.