Given a presentation of a monoid $M$, combined work of Pride and of Guba and Sapir provides an exact sequence connecting the relation bimodule of the presentation (in the sense of Ivanov) with the first homology of the Squier complex of the presentation, which is naturally a $\mathbb ZM$-bimodule. This exact sequence was used by Kobayashi and Otto to prove the equivalence of Pride’s finite homological type property with the homological finiteness condition bi-$\mathrm {FP}_3$. Guba and Sapir used this exact sequence to describe the abelianization of a diagram group. We prove here a generalization of this exact sequence of bimodules for presentations of associative algebras. Our proof is more elementary than the original proof for the special case of monoids.

We prove that a homomorphism between free groups of finite rank equipped with the bi-invariant word metrics associated with finite generating sets is a quasi-isometry if and only if it is an isomorphism.

We present a solution to the conjugacy problem in the group of outer automorphisms of $F_3$, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of essential simple closed curves in the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. In this paper, based on their argument, we prove that the automorphism group of the fine curve graph of a closed nonorientable surface $N$ of genus $g \geq 4$ is isomorphic to the homeomorphism group of $N$.

Let $\Gamma $ be a finitely generated group of matrices over $\mathbb {C}$. We construct an isometric action of $\Gamma $ on a complete $\mathrm {CAT}(0)$ space such that the restriction of this action to any subgroup of $\Gamma $ containing no nontrivial unipotent elements is well behaved. As an application, we show that if M is a graph manifold that does not admit a nonpositively curved Riemannian metric, then any finite-dimensional $\mathbb {C}$-linear representation of $\pi _1(M)$ maps a nontrivial element of $\pi _1(M)$ to a unipotent matrix. In particular, the fundamental groups of such 3-manifolds do not admit any faithful finite-dimensional unitary representations.

Let $H\le F$ be two finitely generated free groups. Given $g\in F$, we study the ideal $\mathfrak I_g$ of equations for g with coefficients in H, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in F. The ideal $\mathfrak I_g$ is a normal subgroup of $H*\langle x\rangle$, and it’s possible to algorithmically compute a finite normal generating set for $\mathfrak I_g$; we give a description of one such algorithm, based on Stallings folding operations. We provide an algorithm to find an equation in w(x)\in$\mathfrak I_g$ with minimum degree, i.e.  such that its cyclic reduction contains the minimum possible number of occurrences of x and x−1; this answers a question of A. Rosenmann and E. Ventura. More generally, we show how to algorithmically compute the set Dg of all integers d such that $\mathfrak I_g$ contains equations of degree d; we show that Dg coincides, up to a finite set, with either $\mathbb N$ or $2\mathbb N$. Finally, we provide examples to illustrate the techniques introduced in this paper. We discuss the case where ${\text{rank}}(H)=1$. We prove that both kinds of sets Dg can actually occur. We show that the equations of minimum possible degree aren’t in general enough to generate the whole ideal $\mathfrak I_g$ as a normal subgroup.

We show that when a finitely presented Bestvina–Brady group splits as an amalgamated product over a subgroup $H$, its defining graph contains an induced separating subgraph whose associated Bestvina–Brady group is contained in a conjugate of $H$.

Given a group G acting faithfully on a set S, we characterize precisely when the twisted Brin–Thompson group SVG is finitely presented. The answer is that SVG is finitely presented if and only if we have the following: G is finitely presented, the action of G on S has finitely many orbits of two-element subsets of S, and the stabilizer in G of any element of S is finitely generated. Since twisted Brin–Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone–Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest.

To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism’s dominating exponential dynamics.

As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn–Handel’s recognition theorem for atoroidal outer automorphisms.

We study the planar 3-colorablesubgroup $\mathcal{E}$ of Thompson’s group F and its even part ${\mathcal{E}_{\rm EVEN}}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of F isomorphic to F, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part ${\mathcal{E}_{\rm EVEN}}$ of the planar 3-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence ${\mathcal{E}_{\rm EVEN}}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with ${\mathcal{E}_{\rm EVEN}}$: two are shown to be irreducible and one to be reducible.

In this paper, we study random walks on groups that contain superlinear-divergent geodesics, in the line of thoughts of Goldsborough and Sisto. The existence of a superlinear-divergent geodesic is a quasi-isometry invariant which allows us to execute Gouëzel’s pivoting technique. We develop the theory of superlinear divergence and establish a central limit theorem for random walks on these groups.

We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.

We investigate when a group of the form $G\times \mathbb {Z}^m\ (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\mathrm {FGFP_a}$), by using the BNS-invariant, and provide some partial answers and nontrivial examples.

We study actions of higher rank lattices $\Gamma <G$ on hyperbolic spaces and we show that all such actions satisfying mild properties come from the rank-one factors of G. In particular, all non-elementary isometric actions on an unbounded hyperbolic space are of this type.

We introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$. A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Röver’s group $V(\Gamma )$, and others of Nekrashevych’s ‘simple groups of dynamical origin’).

Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $, we prove that any $\Gamma $-Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $. More generally, we prove that for a Zariski dense $\theta $-transverse subgroup $\Gamma <G$, any $(\Gamma , \psi )$-Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $, provided the $\psi $-Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

We show that a virtually residually finite rationally solvable (RFRS) group $G$ of type $\mathtt {FP}_n(\mathbb {Q})$ virtually algebraically fibres with kernel of type $\mathtt {FP}_n(\mathbb {Q})$ if and only if the first $n$ $\ell ^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$ are virtually Abelian. It then follows that if $G$ is a virtually RFRS group of type $\mathtt {FP}(\mathbb {Q})$ such that $\mathbb {Z} G$ is Noetherian, then $G$ is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$, which includes (for instance) the class of virtually compact special groups.

We construct the first examples of infinite sharply 2-transitive groups which are finitely generated. Moreover, we construct such a group that has Kazhdan property (T), is simple, has exactly four conjugacy classes and we show that this number is as small as possible.