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

The conjugacy problem in extensions of Thompson's group F

The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-016-1403-9We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Gonçalves in [4] showing that F has property R8, and which can be extended to show that Thompson’s group T also has property R8.Peer ReviewedPostprint (author's final draft

On endomorphisms of torsion-free hyperbolic groups

Postprint (author’s final draft

Dependence and algebraicity over subgroups of free groups

An element $g$ of a free group $F$ depends on a subgroup $H < F$, or is algebraic over $H$, if it satisfies a univariate equation over $H$. Equivalently, when $H$ is finitely-generated, the rank of \gen{H,g} is at most the rank of $H$. We study the elements that depend on subgroups of $F$, the subgroups they generate and the equations that they satisfy.Comment: 11 page

A recursive presentation for Mihailova's subgroup

We give an explicit recursive presentation for Mihailova's sub- group M(H) of Fn Fn corresponding to a nite, concise and Pei er aspherical presentation H = hx1; : : : ; xn jR1; : : : ;Rmi. This partially answers a question of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a nitely generated recursively presented orbit undecidable subgroup of Aut(F3)Postprint (published version

A description of auto-fixed subgroups in a free group

Let F be a finitely generated free group. By using Bestvina-Handel theory, as well as some further improvements, the eigengroups of a given automorphism of F (and its fixed subgroup among them) are globally analyzed and described. In particular, an explicit description of all subgroups of F which occur as the fixed subgroup of some automorphism is given.Peer ReviewedPostprint (author’s final draft

Degrees of compression and inertia for free-abelian times free groups

© 2020. ElsevierWe introduce the concepts of degree of inertia, diG(H), and degree of compression,dcG(H), of a finitely generated subgroupHof a given groupG. For the case of direct productsof free-abelian and free groups, we compute the degree of compression and give an upper boundfor the degree of inertia. Imposing some technical assumptions to the supremum involved in thedefinition of degree of inertia, we introduce the notion called restricted degree of inertia, di'G(H),and, again for the caseZm×Fn, we provide an explicit formula relating it to the restricted degreeof inertia of its projection to the free part, di'Fn(Hp).Both authors are partially supported by the Spanish Agencia Estatalde Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also by the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445)Peer ReviewedPostprint (author's final draft

On automorphism-fixed subgroups of a free group

Let F be a flnitely generated free group, and let n denote its rank. A subgroup H of F is said to be automorphism-flxed, or auto-flxed for short, if there exists a set S of automorphisms of F such that H is precisely the set of elements flxed by every element of S; similarly, H is 1-auto-flxed if there exists a single automorphism of F whose set of flxed elements is precisely H. We show that each auto-flxed subgroup of F is a free factor of a 1-auto-flxed subgroup of F. We show also that if (and only if) n ‚ 3, then there exist free factors of 1-auto-flxed subgroups of F which are not auto-flxed subgroups of F. A 1-auto-flxed subgroup H of F has rank at most n, by the Bestvina-Handel Theorem, and if H has rank exactly n, then H is said to be a maximum-rank 1-auto-flxed subgroup of F, and similarly for auto-flxed subgroups. Hence a maximum-rank auto-flxed subgroup of F is a (maximum-rank) 1-auto-flxed subgroup of F. We further prove that if H is a maximum-rank 1-auto-flxed subgroup of F, then the group of automorphisms of F which flx every element of H is free abelian of rank at most n ¡ 1. All of our results apply also to endomorphisms.Peer ReviewedPostprint (author’s final draft

Stallings automata for free-times-abelian groups: intersections and index

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case. This approach provides a neat geometric description of (even non-(finitely generated)) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals, respectively.The first author was partially supported by CMUP (UID/MAT/ 00144/2019), which is funded by FCT (Portugal) with national (MCTES) and European structural funds through the FEDER programs, under the partnership agreement PT2020. Parts of this project were developed during the participation of the first author in the “Logic and Algorithms in Group Theory” meeting held in the Hausdorff Research Institute for Mathematics (Bonn) in fall 2018. He was also partially supported by MINECO grant PID2019-107444GA-I00 and the Basque Government grant IT974-16. Both authors acknowledge partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also from the Barcelona Graduate School of Mathematics through the “María de Maeztu” Program for Units of Excellence in R&D (MDM-2014-0445).Peer ReviewedPostprint (published version

Twisted conjugacy in braid groups

In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids u, v ∈ Bn and an automorphism ϕ ∈ Aut(Bn), decides whether v = (ϕ(x))−1ux for some x ∈ Bn. As a corollary, we deduce that each group of the form Bn o H, a semidirect product of the braid group Bn by a torsion-free hyperbolic group H, has solvable conjugacy problem.Ministerio de Educación y CienciaJunta de AndalucíaFondo Europeo de Desarrollo Regiona