Related papers: McCool groups of toral relatively hyperbolic group…

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1-->…

Two automorphisms of a simple stable AF algebra with a finite dimensional lattice of lower semicontinuous traces are shown to be outer conjugate if they act in the same way on the K-group and the extremal traces are scaled by numbers which…

Given isometric actions by a group G on finitely many \delta-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for each action. As an application we prove a…

The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and…

The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups.…

Given two automorphisms of a group $G$, one is interested in knowing whether they are conjugate in the automorphism group of $G$, or in the abstract commensurator of $G$, and how these two properties may differ. When $G$ is the fundamental…

Given a connected large-type Artin group $A_\Gamma$, we introduce a deformation space $\mathcal{D}$. If $\Gamma$ is triangle-free, or has all labels at least 6, we show that this space is canonical, in that it depends only on the…

We initiate the study of outer automorphism groups of special groups $G$, in the Haglund-Wise sense. We show that ${\rm Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an…

Let K be a fine hyperbolic graph and G be a group acting on K with finite quotient. We prove that G is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups…

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all…

We investigate the action of the automorphism group of an acylindrically hyperbolic group G on its space of homogeneous quasimorphisms, and identify its kernel with the subgroup of "strongly commensurating" automorphisms. We deduce that if…

Let ${\cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected \'etale cover ${\cal M}^\l$ of ${\cal…

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is,…

Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show that virtually free groups have a…

We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ induces an isomorphism between the homology groups…

Let $G$ be a torsion-free hyperbolic group and $\alpha$ an automorphism of $G$. We show that there exists a canonical collection of subgroups that are polynomially growing under $\alpha$, and that the mapping torus of $G$ by $\alpha$ is…

For a group $G$ of finite Kurosh rank and for some arbiratily free product decomposition of $G$, $G = H_1 \ast H_2 \ast ... \ast H_r \ast F_q$, where $F_q$ is a finitely generated free group, we can associate some (relative) outer space…

For every Cuntz--Krieger groupoid, we show that there is a topologically free boundary action of the outer automorphism group of its topological full group on the Hilbert cube. In particular, these outer automorphism groups, including the…

Let $H$ be a torsion-free $\delta$-hyperbolic group with respect to a finite generating set $S$. Let $a_1,..., a_n$ and $a_{1*},..., a_{n*}$ be elements of $H$ such that $a_{i*}$ is conjugate to $a_i$ for each $i=1,..., n$. Then, there is a…

Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au- tomorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and…