Subgroups of even Artin groups of FC-type

We prove a Tits alternative theorem for subgroups of finitely generated even Artin groups of FC type (EAFC groups), stating that there exists a finite index subgroup such that every subgroup of it is either finitely generated abelian, or maps onto a non-abelian free group. Parabolic subgroups play a key role, and we show that parabolic subgroups of EAFC groups are closed under taking roots.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasFALSEMinisterio de Ciencia e InnovaciónSantander-UCMunpu

On the classifying space for proper actions of groups with cyclic torsion

In this paper we introduce a common framework for describing the topological part of the Baum-Connes conjecture for a wide class of groups. We compute the Bredon homology for groups with aspherical presentation, one-relator quotients of products of locally indicable groups, extensions of Zn by cyclic groups, and fuchsian groups. We take advantage of the torsion structure of these groups to use appropriate models of the universal space for proper actions which allow us, in turn, to extend some technology defined by Mislin in the case of one-relator groups.Ministerio de Ciencia e InnovaciónEngineering and Physical Sciences Research Counci

Intersection of parabolic subgroups in even Artin groups of FC-type

We show that the intersection of parabolic subgroups of an even finitely generated FC-type Artin group is again a parabolic subgroup

Parabolic subgroups acting on the additional length graph

Let A ≠ A1;A2;I2m be an irreducible Artin–Tits group of spherical type. We show that the periodic elements of A and the elements preserving some parabolic subgroup of A act elliptically on the additional length graph CAL(A), a hyperbolic, infinite diameter graph associated to A constructed by Calvez and Wiest to show that A/Z(A) is acylindrically hyperbolic. We use these results to find an element g ∈ A such that ≅ P * for every proper standard parabolic subgroup P of A. The length of g is uniformly bounded with respect to the Garside generators, independently of A. This allows us to show that, in contrast with the Artin generators case, the sequence ω(An,S)(with n ∈ N) of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity

Geodesic Growth of some 3-dimensional RACGs

We give explicit formulas for the geodesic growth series of a Right Angled Coxeter Group (RACG) based on a link-regular graph that is 4-clique free, i.e. without tetrahedrons

The Hanna Neumann conjecture for surface groups

The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the Strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition we show that a retract in a surface group is inert. It is know that this implies the Dicks-Ventura inert conjecture for free groups

On the asymptotics of visible elements and homogeneous equations in surface groups

Let F be a group whose abelianization is Zk, k 2. An element of F is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity