On Cayley graphs of virtually free groups

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them

Tits alternatives for graph products

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products

Commensurating endomorphisms of acylindrically hyperbolic groups and applications

We prove that the outer automorphism group $Out(G)$ is residually finite when the group $G$ is virtually compact special (in the sense of Haglund and Wise) or when $G$ is isomorphic to the fundamental group of some compact $3$-manifold. To prove these results we characterize commensurating endomorphisms of acylindrically hyperbolic groups. An endomorphism $\phi$ of a group $G$ is said to be commensurating, if for every $g \in G$ some non-zero power of $\phi(g)$ is conjugate to a non-zero power of $g$. Given an acylindrically hyperbolic group $G$, we show that any commensurating endomorphism of $G$ is inner modulo a small perturbation. This generalizes a theorem of Minasyan and Osin, which provided a similar statement in the case when $G$ is relatively hyperbolic. We then use this result to study pointwise inner and normal endomorphisms of acylindrically hyperbolic groups.Comment: 47 pages. v4: final version, to appear in Groups, Geometry and Dynamic

Degree of commutativity of infinite groups

First published in Proceedings of the American Mathematical Society in volum 145, number 2, 2016, published by the American Mathematical SocietyWe prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups, where the hypothesis of residual finiteness is always satisfied). We also show that, for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.Peer ReviewedPostprint (author's final draft

Non-orientable surface-plus-one-relation groups

Recently Dicks–Linnell determined the L2-Betti numbers of the orientable surface-plus-one-relation groups, and their arguments involved some results that were obtained topologically by Hempel and Howie. Using algebraic arguments, we now extend all these results of Hempel and Howie to a larger class of two-relator groups, and we then apply the extended results to determine the L2-Betti numbers of the non-orientable surface-plus-one-relation group

Degree of commutativity of infinite groups

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups). We also show that for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero

Kurosh rank of intersections of subgroups of free products of right-orderable groups

We prove that the reduced Kurosh rank of the intersection of two subgroups H and K of a free product of right-orderable groups is bounded above by the product of the reduced Kurosh ranks of H and K.In particular, taking the fundamental group of a graph of groups with trivial vertex and edge groups, and its Bass-Serre tree, our theorem becomes the desired inequality of the usual strengthened Hanna Neumann conjecture for free groups

Complexity of positive cones of limit groups

A left-order on a group G is totally determined by its positive cone P, that is the elements in G that are greater than the identity. The set P is a subsemigroup and we can ask ourselves when this semigroup can befinitely generated, described by a regular language,etc. A recent resultof S.Hermiller and Z.Sunic shows that non-abelian free groups do not have regular positivecones. I will discuss this and how to generalize this result to limit groups. The talk will be based on joint work with C.Rivas and I will report some results of my PhD student H.L.Su. (Yago Antolin is from the Unversidad Autónoma de Madrid)Non UBCUnreviewedAuthor affiliation: Universidad Autónoma de MadridResearche

On Cayley graphs of virtually free groups

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them<br/