Related papers: Limit groups and groups acting freely on $\bbR^n$-…

It was recently proven that all free and many virtually free verbally closed subgroups are algebraically closed in any group. We establish sufficient conditions for a group that is an extension of a free non-abelian group by a group…

We prove that, given a finitely generated subgroup $H$ of a free group $F$, the following questions are decidable: is $H$ closed (dense) in $F$ for the pro-(met)abelian topology? is the closure of $H$ in $F$ for the pro-(met)abelian…

Gromov asked what a typical (finitely presented) group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely generated group is known to share some important properties with the…

We start by studying the distribution of (cyclically reduced) elements of the free groups with respect to their abelianization. We derive an explicit generating function, and a limiting distribution, by means of certain results (of…

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order…

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the…

It is shown that a finitely generated branch group has Serre's property (FA) if and only if it does not surject onto the infinite cyclic group or the infinite dihedral group. An example of a finitely generated self-similar branch group…

We construct a new family of examples of parabolically geometrically finite subgroups of the mapping class group in the sense of Dowdall-Durham-Leininger-Sisto and prove they are undistorted in Mod($S$).

As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding…

In this article we study the K- and L-theory of groups acting on trees. We consider the problem in the context of the fibered isomorphism conjecture of Farrell and Jones. We show that in the class of residually finite groups it is enough to…

In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group $H \in…

Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental…

We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring,…

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our…

We consider the class of countable groups possessing an action on a finite product of hyperbolic graphs where every infinite order element acts loxodromically. When the graphs are locally finite, we obtain strong structure theorems for the…

We show that every finitely generated free-by-cyclic group $G$ admits a largest acylindrical action on a hyperbolic space $X$ obtained by coning off maximal product subgroups of $G$. We characterise Morse geodesics of $G$ as those that…

We say that a finitely generated group $\Gamma$ is self-simulable if every effectively closed action of $\Gamma$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $\Gamma$-subshift of finite type.…

We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in…

Serre in "Trees" laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree.…

An approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented. Several results related to the whole class are formulated. In particular, all finite,…