arXiv subjects

Eduardo Martinez-Pedroza

Publications and source records attributed to Eduardo Martinez-Pedroza.

13 recordsLinked to original sources

arXiv2026

Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups

In this article, we prove that, given two finite connected graphs $Γ_1$ and $Γ_2$, if the two right-angled Artin groups $A(Γ_1)$ and $A(Γ_2)$ are quasi-isometric, then the infinite pointed sums $\bigvee_\mathbb{N} Γ_1^{\bowtie}$ and $\bigvee_\mathbb{N} Γ_2^{\bowtie}$ are homotopy equivalent, where $Γ_i^{\bowtie}$ denotes the simplicial complex whose vertex-set is $Γ_i$ and whose simplices are given by joins. These invariants are extracted from a study, of independent interest, of the homotopy types of several complexes of hyperplanes in quasi-median graphs (such as one-skeleta of CAT(0) cube complexes). For instance, given a quasi-median graph $X$, the \emph{crossing complex} $\mathrm{Cross}^\triangle(X)$ is the simplicial complex whose vertices are the hyperplanes (or $θ$-classes) of $X$ and whose simplices are collections of pairwise transverse hyperplanes. When $X$ has no cut-vertex, we show that $\mathrm{Cross}^\triangle(X)$ is homotopy equivalent to the pointed sum of the links of all the vertices in the prism-completion $X^\square$ of $X$.

math.GR↗

arXiv2021

Local Quasiconvexity of Groups acting on Small Cancellation Complexes

Given a group acting cellularly and cocompactly on a simply-connected 2-complex, we provide a criterion establishing that all finitely generated subgroups have quasiconvex orbits. This work generalizes the "perimeter method". As an application, we show that high-powered one-relator products $A \ast B / \nclose{r^n}$ are coherent if $A$ and $B$ are coherent.

math.GR↗

arXiv2017

Subgroups of Relatively Hyperbolic Groups of Bredon Cohomological Dimension 2

A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension $2$ is closed under taking finitely presented (or more generally $FP_2$) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension $2$ with respect to the family of parabolic subgroups. A class of groups where our result applies consists of $C'(1/6)$ small cancellation products. The proof relies on an algebraic approach to relative homological Dehn functions, and a characterization of relative hyperbolicity in the framework of finiteness properties over Bredon modules and homological Isoperimetric inequalities.

math.GR↗

arXiv2017

The Coarse Geometry of Hartnell's Firefighter Problem on Infinite Graphs

In this article, we study Hartnell's Firefighter Problem through the group theoretic notions of growth and quasi-isometry. A graph has the $n$-containment property if for every finite initial fire, there is a strategy to contain the fire by protecting $n$ vertices at each turn. A graph has the constant containment property if there is an integer $n$ such that it has the $n$-containment property. Our first result is that any locally finite connected graph with quadratic growth has the constant containment property; the converse does not hold. This result provides a unified way to recover previous results in the literature, in particular the class of graphs satisfying the constant containment property is infinite. A second result is that in the class of graphs with bounded degree, having the constant containment property is preserved by quasi-isometry. Some sample consequences of the second result are that any regular tiling of the Euclidean plane has the fire containment property; no regular tiling of the $n$-dimensional Euclidean space has the containment property if $n>2$; and no regular tiling of the $n$-dimensional hyperbolic space has the containment property if $n\geq 2$. We prove analogous results for the $\{f_n\}$-containment property, where $f_n$ is an integer sequence corresponding to the number of vertices protected at time $n$. In particular, we positively answer a conjecture by Develin and Hartke by proving that the $d$-dimensional square grid $\mathbb{L}^d$ does not satisfy the $cn^{d-3}$-containment property for any constant $c$.

math.CO↗

arXiv2017

Dismantlable classifying space for the family of parabolic subgroups of a relatively hyperbolic group

Let $G$ be a group hyperbolic relative to a finite collection of subgroups $\mathcal P$. Let $\mathcal F$ be the family of subgroups consisting of all the conjugates of subgroups in $\mathcal P$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{\mathcal F} G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.

math.GR↗

arXiv2015

A note on fine graphs and homological isoperimetric inequalities

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers.

math.GT↗

arXiv2015

A Subgroup Theorem for Homological Filling Functions

We use algebraic techniques to study homological filling functions of groups and their subgroups. If $G$ is a group admitting a finite $(n+1)$--dimensional $K(G,1)$ and $H \leq G$ is of type $F_{n+1}$, then the $n^{th}$--homological filling function of $H$ is bounded above by that of $G$. This contrast with known examples where such inequality does not hold under weaker conditions on the ambient group $G$ or the subgroup $H$. We include applications to hyperbolic groups and homotopical filling functions.

math.GR↗

arXiv2014

Lifting Group Actions, Equivariant Towers and Subgroups of Non-positively Curved Groups

If $\mathcal C$ is a class of complexes closed under taking full subcomplexes and covers and $\mathcal G$ is the class of groups admitting proper and cocompact actions on one-connected complexes in $\mathcal C$, then $\mathcal G$ is closed under taking finitely presented subgroups. As a consequence the following classes of groups are closed under taking finitely presented subgroups: groups acting geometrically on regular $CAT(0)$ simplicial complexes of dimension $3$, $k$-systolic groups for $k\geq 6$, and groups acting geometrically on $2$-dimensional negatively curved complexes. We also show that there is a finite non-positively curved cubical $3$-complex which is not homotopy equivalent to a finite non-positively curved regular simplicial $3$-complex. We included other applications to relatively hyperbolic groups and diagramatically reducible groups. The main result is obtained by developing a notion of equivariant towers which is of independent interest.

math.GR↗

arXiv2012

Virtual Amalgamation of Relatively Quasiconvex Subgroups

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.

math.GT↗

arXiv2011

On Quasiconvexity and Relative Hyperbolic Structures

Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when quasicovexity relative to A implies quasiconvexity relative to B. We also show that quasiconvexity relative to B implies quasiconvexity relative to A. Some applications are discussed.

math.GR↗

arXiv2010

Relative Quasiconvexity using Fine Hyperbolic Graphs

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to quasiconvexity generalizes the other definitions in the literature that apply only for countable relatively hyperbolic groups. We also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic.

math.GR↗

arXiv2010

Separation of Relatively Quasiconvex Subgroups

Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex subgroup of a relatively hyperbolic group G to the separability of a quasiconvex subgroup of a hyperbolic quotient G/N. A result of Agol, Groves, and Manning is then applied.

math.GR↗

arXiv2008

Combination of quasiconvex subgroups of relatively hyperbolic groups

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two quasiconvex subgroups $Q$ and $R$ is quasiconvex and isomorphic to $Q \ast_{Q\cap R} R$. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented.

math.GR↗