Related papers: Self-simulable groups

We construct the first example of a finitely generated group which has Serre's property (FA) (i.e., whenever it acts on a simplicial tree it fixes a vertex), but admits a fixed point-free action on an $\mathbb{R}$-tree with finite arc…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

We introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $\Gamma$ be a computable amenable group, then there always exists a canonically computable tempered two-sided…

For a non-empty class of groups $\cal L$, a finite group $G = AB$ is said to be an $\cal L$-connected product of the subgroups $A$ and $B$ if $\langle a, b\rangle \in \cal L$ for all $a \in A$ and $b \in B$. In a previous paper, we prove…

Given an arbitrary, finitely presented, residually finite group $\Gamma$, one can construct a finitely generated, residually finite, free-by-free group $M_\Gamma = F_\infty\rtimes F_4$ and an embedding $M_\Gamma \hookrightarrow (F_4\ast…

The set of finitely generated subgroups of the group $PL_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R.~Thompson's group $F$. In this paper we show that…

We say that a subset $X$ quasi-isometrically boundedly generates a finitely generated group $\Gamma$ if each element $\gamma$ of a finite-index subgroup of $\Gamma$ can be written as a product $\gamma = x_1 x_2 \cdots x_r$ of a bounded…

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is…

We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of…

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We have also given…

A pseudomodular group is a discrete subgroup $\Gamma \leq PGL(2,\mathbb{Q})$ which is not commensurable with $PSL(2,\mathbb{Z})$ and has cusp set precisely $\mathbb{Q}\cup\{\infty\}$. The existence of such groups was proved by Long and…

Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $\Gamma$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely,…

We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism…

If $\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group…

We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that…

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.)…

A finitely generated solvable group with unbounded iterated identity is constructed.

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is…

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for…

We construct examples of finitely generated groups L that have non-trivial actions on $\mathbb{R}$-trees but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto L does have…