Abstract. 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.

Abstract. Given a finitely generated relatively hyperbolic group G, we construct a finite generating set X of G such that (G, X) has the 'falsification by fellow traveler property' provided that the parabolic subgroups {Hω}ω∈Ω have this property with respect to the generating sets {X ∩ Hω}ω∈Ω. This implies that groups hyperbolic relative to virtually abelian subgroups, which include all limit groups and groups acting freely on R n -trees, or geometrically finite hyperbolic groups, have generating sets for which the language of geodesics is regular, and the complete growth series and complete geodesic series are rational. As an application of our techniques, we prove that if each Hω admits a geodesic biautomatic structure over X ∩ Hω, then G has a geodesic biautomatic structure.Similarly, we construct a finite generating set X of G such that (G, X) has the 'bounded conjugacy diagrams' property or the 'neighbouring shorter conjugate' property if the parabolic subgroups {Hω}ω∈Ω have this property with respect to the generating sets {X ∩Hω}ω∈Ω. This implies that a group hyperbolic relative to abelian subgroups has a generating set for which its Cayley graph has bounded conjugacy diagrams, a fact we use to give a cubic time algorithm to solve the conjugacy problem. Another corollary of our results is that groups hyperbolic relative to virtually abelian subgroups have a regular language of conjugacy geodesics.

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 ϕ of a group G is said to be commensurating, if for every g ∈ G some non-zero power of ϕ(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.

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, 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.

Abstract. Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes.We then explore Rivin's conjecture for finitely generated acylindrically hyperbolic groups and prove a formal language version of it, namely that no set of minimal length conjugacy representatives can be unambiguous context-free.2010 Mathematics Subject Classification: 20F67, 68Q45.