Related papers: Completely Metrisable Groups Acting on Trees

Reid-Smith recently parametrised groups acting on trees with Tits' independence property (P) using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally…

A finitely generated group $G$ acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag--Solitar group ($GBS$ group). We prove that a 1-knot group $G$ is $GBS$ group iff $G$ is a torus-knot group…

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal…

We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R^n-trees. We first prove that Sela's limit groups do have a free action on an R^n-tree. We then prove that a finitely generated group having…

We show that the semidirect product of a group $C$ by $A*_D B$ is isomorphic to the free product of $A\rtimes C$ and $B\rtimes C$ amalgamated at $D\rtimes C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to…

We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras,…

We examine the question of which finitely generated groups act properly on a finite product of simplicial trees, considering both arbitrary trees and where all trees are locally finite. In the second case we present evidence in favour of…

We give necessary and sufficient conditions under which a quasi-action of any group on an arbitrary metric space can be reduced to a cobounded isometric action on some bounded valence tree, following a result of Mosher, Sageev and Whyte.…

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of…

This article studies a structural aspect of measure-preserving actions of products of countable discrete groups, involving a so-called 'synergodic decomposition' in terms of the ergodic components of the actions of the two factor groups. We…

A well known theorem of Burns and Romanovskii states that a free product of subgroup separable groups is itself subgroup separable. We provide a proof using the language of immersions and coverings of graphs of groups, due to Bass.

Given a bounded valence, bushy tree T, we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T'. This theorem has many applications: quasi-isometric rigidity…

We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal…

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if…

This paper explores acylindrical actions on trees, building on previous works related to the mapping class group and projection complexes. We demonstrate that the quotient action of a $1$-acylindrical action of a group on a tree by an…

We prove an accessibility theorem for finite-index splittings of groups. Given a finitely presented group G there is a number n(G) such that, for every reduced locally finite G-tree T with finitely generated stabilizers, T/G has at most…

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the…

We introduce two families of examples of groups acting on trees, one consisting of group amalgamations and the other consisting of HNN-extensions, motivated by the problems of $C^*$-simplicity and unique trace property. Moreover, we prove…

In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $\mathcal{F}$-amenable" group action, where $\mathcal{F}$ is a family of subgroups. We show how a finitely $\mathcal{F}$-amenable action of a…

We examine the question of which finitely generated groups act properly on a finite product of locally finite simplicial trees and present evidence in favour of hyperbolic surface groups having such an action. We also give a completely…