We show that every effectively closed action of a finitely generated group G over a Cantor set can be obtained as a topological factor of the G-subaction of a (G × H1 × H2)-subshift of finite type for any choice of infinite and finitely generated groups H1, H2. As a consequence, we obtain that every group of the form G1 ×G2 ×G3 admits a non-empty strongly aperiodic SFT subject to the condition ...

For many groups the structure of finitely generated subgroups is generically simple. That is with asymptotic density equal to one a randomly chosen finitely generated subgroup has a particular well-known and easily analyzed structure. For example a result of D. B. A. Epstein says that a finitely generated subgroup of GL(n,R) is generically a free group. We say that a group G has the generic fre...

We prove that a finitely generated soluble residually finite group has polynomial index growth if and only if it is a minimax group. We also show that if a finitely generated group with PIG is residually finite-soluble then it is a linear group. These results apply in particular to boundedly generated groups; they imply that every infinite BG residually finite group has an infinite linear quoti...

LET G be a residually-finite torsion-free group. Is Gthe profinite completion of G-torsion free? This question was asked in [CKL] where it was shown that if G is a finitely generated metabelian-by-finite group then indeed G is torsion free. On the other hand Evans [E] showed that if G is not finitely generated then it is possible that G has torsion. His example is also metabelian. In this note ...

In [24], confirming a conjecture of Hjorth-Kechris [16], Thomas-Velickovic proved that the isomorphism relation on the space Gfg of finitely generated groups is a universal countable Borel equivalence relation. (Here Gfg denotes the Polish space of finitely generated groups introduced by Grigorchuk [12]; i.e. the elements of Gfg are the isomorphism types of marked groups (G, c ), where G is a f...

Stallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups. Here we adapt the method to the class of finitely generated groups using Sageev’s generalisation of Bass Serre theory concerning group p...

A group has homological dimension ≤ 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L -Betti numbers. We also show that a finitely generated elementary amenable group G of cohomological dimension ≤ 2 possesses a finite 2dimensional model for BG and in particular that G is finitely presented and the trivial ZG-module Z has a 2-dimensional reso...

We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable gr...

We study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by L. Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. ...

We prove that a finitely generated group G hyperbolic relative to the collection of finitely generated subgroups {H1, . . . , Hm} has the Rapid Decay property if and only if each Hi , i = 1, 2, . . . ,m, has the Rapid Decay property.