arXiv subjects
Publications and source records attributed to Eduardo Martínez-Pedroza.
Let $H$ be a group acting on a simply-connected diagrammatically reducible combinatorial 2-complex $X$ with fine 1-skeleton. If the fixed point set $X^ H$ is non-empty, then it is contractible. Having fine 1-skeleton is a weaker version of being locally finite.
In this note, we study the equivalence of Morse and stable subgroups in the framework of the coset intersection complex. Under certain conditions on a coset intersection complex of a group, we prove that infinite-index Morse subgroups are stable.
We show that there is a collection of subgroups of the mapping class group of a surface such that the associated coset intersection complex is quasi-isometric and homotopy equivalent to the curve complex. Moreover, we prove that these two complexes are combinatorially equivalent in the sense that one can be obtained from the other via taking a nerve. As an application, we prove that the automorphism group of this coset intersection complex is the extended mapping class group, a result in the spirit of for Ivanov's metaconjecture.
We address a question from \cite{BKV25} regarding the finiteness of the homological $R$-isoperimetric function. Let $R$ be a subfield of the complex numbers $\mathbb{C}$ with the absolute value norm. We prove that for any group $G$ that admits a finite $(n+1)$-dimensional model for $K(G,1)$, the homological $n$-isoperimetric function of $G$ over $R$ is either linear or takes infinite values. In particular, by results of Gersten and Mineyev, in the class of groups admitting a finite $2$-dimensional classifying space, the homological $1$-dimensional isoperimetric function over $R$ only captures hyperbolicity. This follows as a particular case of a more general result proved in this note.
Given a pointed metric space $(X,\mathsf{dist}, w)$ on $n$ points, its Gromov's approximating tree is a 0-hyperbolic pseudo-metric space $(X,\mathsf{dist}_T)$ such that $\mathsf{dist}(x,w)=\mathsf{dist}_T(x,w)$ and $\mathsf{dist}(x, y)-2 δ\log_2n \leq \mathsf{dist}_T (x, y) \leq \mathsf{dist}(x, y)$ for all $x, y \in X$ where $δ$ is the Gromov hyperbolicity of $X$. On the other hand, the all pairs bottleneck paths (APBP) problem asks, given an undirected graph with some capacities on its edges, to find the maximal path capacity between each pair of vertices. In this note, we prove: $\bullet$ Computing Gromov's approximating tree for a metric space with $n+1$ points from its matrix of distances reduces to solving the APBP problem on an connected graph with $n$ vertices. $\bullet$ There is an explicit algorithm that computes Gromov's approximating tree for a graph from its adjacency matrix in quadratic time. $\bullet$ Solving the APBP problem on a weighted graph with $n$ vertices reduces to finding Gromov's approximating tree for a metric space with $n+1$ points from its distance matrix.
The first author and Oguni introduced a class of groups of non-positive curvature, called coarsely convex group. The recent success of the theory of groups which are hyperbolic relative to a collection of subgroups has motivated the study of other properties of groups from the relative perspective. In this article, we propose definitions for the notions of weakly semihyperbolic, semihyperbolic, and coarsely convex group pairs extending the corresponding notions in the non-relative case. The main result of this article is the following combination theorem. Let $\mathcal{A}_{wsh}$, $\mathcal{A}_{sh}$, and $\mathcal{A}_{gcc}$ denote the classes of group pairs that are weakly semihyperbolic, semihyperbolic, and geodesic coarsely convex respectively. Let $\mathcal A$ be one of the classes $\mathcal{A}_{wsh}$, $\mathcal{A}_{sh}$, and $\mathcal{A}_{gcc}$. Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a non-trivial finite collection of subgroups $\mathcal{H}$ of $G$ satisfying the following properties. If each $(G_v, \mathcal{H}_v)$ is in $\mathcal{A} $, then $(G,\mathcal{H})$ is in $\mathcal{A}$. The main results of the article are combination theorems generalizing results of Alonso and Bridson; and Fukaya and Matsuka.
The weak-cop number of a graph, a variation of the cop number, is an invariant suitable for infinite graphs and is a quasi-isometric invariant. While for any $m\in\mathbb{Z}_+\cup\{\infty\}$ there exist locally finite infinite graphs with weak-cop number $m$, it is an open question whether there exists locally finite vertex transitive graphs whose weak-cop number is different than $1$ and $\infty$. We test this question on Cayley graphs of wreath products, these are objects known for their exotic geometries. We prove that Cayley graphs of wreath products of nontrivial groups by infinite groups have infinite weak-cop number. The result is proved by defining a new pursuit and evasion game and proving the existence of strategies for the evader. We also include a short argument that Cayley graphs of Thompson's group $F$ have infinite weak cop number.
For a pair $(G,\mathcal{P})$ consisting of a group and finite collection of subgroups, we introduce a simplicial $G$-complex $\mathcal{K}(G,\mathcal{P})$ called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of $\mathcal{K}(G,\mathcal{P})$ are quasi-isometric invariants of the group pair $(G,\mathcal{P})$. Classical properties of $\mathcal{P}$ in $G$ correspond to topological or geometric properties of $\mathcal{K}(G,\mathcal{P})$, such as having finite height, having finite width, being almost malnormal, admiting a malnormal core, or having thickness of order one. As applications, we obtain that a number of algebraic properties of $\mathcal{P}$ in $G$ are quasi-isometry invariants of the pair $(G,\mathcal{P})$. For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex $\mathcal{K}(G,\mathcal{P})$ is quasi-isometric to the Extension graph; in particular, it is quasi-isometric to a tree.
Consider the following classes of pairs consisting of a group and a finite collection of subgroups: $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right\}$ and $ \mathcal{D}= \left\{ (G,\mathcal H) \mid \text{the relative Dehn function of $(G,\mathcal H)$ is well-defined} \right\}.$ Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a finite collection of subgroups $\mathcal{H}$ of $G$ such that: $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$, then $(G,\mathcal{H})$ is in $\mathcal C$. $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$, then $(G,\mathcal{H})$ is in $\mathcal D$. $\bullet$ For any vertex $v$ and for any $g\in G_v$, the element $g$ is conjugate to an element in some $Q\in\mathcal{H}_v$ if and only if $g$ is conjugate to an element in some $H\in\mathcal{H}$. That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.
We introduce two variations of the cops and robber game on graphs. These games yield two invariants in $\mathbb{Z}_+\cup\{\infty\}$ for any connected graph $Γ$, the {weak cop number $\mathsf{wcop}(Γ)$} and the {strong cop number $\mathsf{scop}(Γ)$}. These invariants satisfy that $\mathsf{scop}(Γ)\leq\mathsf{wcop}(Γ)$. Any graph that is finite or a tree has strong cop number one. These new invariants are preserved under small local perturbations of the graph, specifically, both the weak and strong cop numbers are quasi-isometric invariants of connected graphs. More generally, we prove that if $Δ$ is a quasi-retract of $Γ$ then $\mathsf{wcop}(Δ)\leq\mathsf{wcop}(Γ)$ and $\mathsf{scop}(Δ)\leq\mathsf{scop}(Γ)$. We exhibit families of examples of graphs with arbitrary weak cop number (resp. strong cop number). We prove that hyperbolic graphs have strong cop number one. We also prove that one-ended non-amenable locally-finite vertex-transitive graphs have infinite weak cop number. We raise the question of whether there exists a connected vertex transitive graph with finite weak (resp. strong) cop number different than one.
The main objects of study in this article are pairs $(G, \mathcal{H})$ where $G$ is a topological group with a compact open subgroup, and $\mathcal{H}$ is a finite collection of open subgroups. We develop geometric techniques to study the notions of $G$ being compactly generated and compactly presented relative to $\mathcal H$. This includes topological characterizations in terms of discrete actions of $G$ on complexes, quasi-isometry invariance of certain graphs associated to the pairs $(G,\mathcal H)$ when $G$ is compactly generated relative to $\mathcal H$, and extensions of known results for the discrete case. For example, generalizing results of Osin for discrete groups, we show that in the case that $G$ is compactly presented relative to $\mathcal H$: $\bullet$ if $G$ is compactly generated, then each subgroup $H\in \mathcal H$ is compactly generated; $\bullet$ if each subgroup $H\in \mathcal H$ is compactly presented, then $G$ is compactly presented. The article also introduces an approach to relative hyperbolicity for pairs $(G, \mathcal H)$ based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, we prove that if $G$ is hyperbolic relative to $\mathcal H$ then $G$ is compactly presented relative to $\mathcal H$. As applications of the results of the article we prove combination results for coherent topological groups with a compact open subgroup, and extend McCammond-Wise perimeter method to this general framework.
Given a non-decreasing function $f \colon \mathbb{N} \to \mathbb{N}$ we define a single player game on (infinite) connected graphs that we call fire retaining. If a graph $G$ admits a winning strategy for any initial configuration (initial fire) then we say that $G$ has the $f$-retaining property; in this case if $f$ is a polynomial of degree $d$, we say that $G$ has the polynomial retaining property of degree $d$. We prove that having the polynomial retaining property of degree $d$ is a quasi-isometry invariant in the class of uniformly locally finite connected graphs. Henceforth, the retaining property defines a quasi-isometric invariant of finitely generated groups. We prove that if a finitely generated group $G$ splits over a quasi-isometrically embedded subgroup of polynomial growth of degree $d$, then $G$ has polynomial retaining property of degree $d-1$. Some connections to other work on quasi-isometry invariants of finitely generated groups are discussed and some questions are raised.
We give technical conditions for a quasi-isometry of pairs to preserve a subgroup being hyperbolically embedded. We consider applications to the quasi-isometry and commensurability invariance of acylindrical hyperbolicity of finitely generated groups.
For a finitely generated group $G$ and collection of subgroups $\mathcal{P}$ we prove that the relative Dehn function of a pair $(G,\mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that for a cocompact simply connected combinatorial $G$-$2$-complex $X$ with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the $1$-skeleton of $X$ is fine. We also show that if $H$ is a hyperbolically embedded subgroup of a finitely presented group $G$, then the relative Dehn function of the pair $(G, H)$ is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group $\mathrm{BS}(k,l)$ has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither $k$ divides $l$ nor $l$ divides $k$.
This survey studies pairs $(G,\mathcal{P})$ with $G$ a finitely generated group and $\mathcal{P}$ a (finite) collection of subgroups of $G$. We explore the notion of quasi-isometry of such pairs and the notion of a qi-characteristic collection of subgroups. Both notions are abstractions of phenomena that have appeared repeatedly in the work of several people within the quasi-isometric rigidity realm.
For a finitely generated group $G$, let $H(G)$ denote Bowditch's taut loop length spectrum. We prove that if $G=(A\ast B) / \langle\!\langle \mathcal R \rangle\!\rangle $ is a $C'(1/12)$ small cancellation quotient of a the free product of finitely generated groups, then $H(G)$ is equivalent to $H(A) \cup H(B)$. We use this result together with bounds for cohomological and geometric dimensions, as well as Bowditch's construction of continuously many non-quasi-isometric $C'(1/6)$ small cancellation $2$-generated groups to obtain our main result: Let $\mathcal{G}$ denote the class of finitely generated groups. The following subclasses contain continuously many one-ended non-quasi-isometric groups: $\bullet\left\{G\in \mathcal{G} \colon \underline{\mathrm{cd}}(G) = 2 \text{ and } \underline{\mathrm{gd}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \underline{\underline{\mathrm{cd}}}(G) = 2 \text{ and } \underline{\underline{\mathrm{gd}}}(G) = 3 \right\}$ $\bullet\left\{G\in \mathcal{G} \colon \mathrm{cd}_{\mathbb{Q}}(G)=2 \text{ and } \mathrm{cd}_{\mathbb{Z}}(G)=3 \right\}$ On our way to proving the aforementioned results, we show that the classes defined above are closed under taking relatively finitely presented $C'(1/12)$ small cancellation quotients of free products, in particular, this produces new examples of groups exhibiting an Eilenberg-Ganea phenomenon for families. We also show that if there is a finitely presented counter-example to the Eilenberg-Ganea conjecture, then there are continuously many finitely generated one-ended non-quasi-isometric counter-examples.
Let $G$ be a group and $H$ a subgroup of $G$. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch's approach to relatively hyperbolic groups in terms of fine graphs.
Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P\leq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,\mathcal{P})$ where $G$ is a finitely generated group and $\mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-characteristic collection. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we list in the article some of them and provide other examples. The first part of the article proves: if $G$ and $H$ are finitely generated quasi-isometric groups and $\mathcal{P}$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $\mathcal{Q}$ of $H$ such that $ (G, \mathcal{P})$ and $(H, \mathcal{Q})$ are quasi-isometric pairs. The second part of the article studies the number of filtered ends $\tilde e (G, P)$ of a pair of groups, a notion introduced by Bowditch, and provides an application of our main result: if $G$ and $H$ are quasi-isometric groups and $P\leq G$ is qi-characterstic, then there is $Q\leq H$ such that $\tilde e (G, P) = \tilde e (H, Q)$.