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G. Christopher Hruska

Publications and source records attributed to G. Christopher Hruska.

At least 19 recordsLinked to original sources
arXiv2024

Relatively hyperbolic groups with planar boundaries

In this article, we prove a version of Martin and Skora's conjecture that convergence groups on the $2$-sphere are covered by Kleinian groups. Given a relatively hyperbolic group pair $(G,\mathcal{P})$ with planar boundary and no Sierpinski carpet or cut points in its boundary, and with $G$ one ended and virtually having no $2$-torsion, we show that $G$ is virtually Kleinian. We also give applications to various versions of the Cannon conjecture and to convergence groups acting on $S^2$.

math.GR↗
arXiv2024

Coarse Alexander duality for pairs and applications

For a group $G$ (of type $F$) acting properly on a coarse Poincaré duality space $X$, Kapovich-Kleiner introduced a coarse version of Alexander duality between $G$ and its complement in $X$. More precisely, the cohomology of $G$ with group ring coefficients is dual to a certain Čech homology group of the family of increasing neighborhoods of a $G$-orbit in $X$. This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse $PD(n)$ spaces. In this paper we introduce a relative version of this Čech homology that satisfies the Eilenberg-Steenrod Exactness Axiom, and we prove a relative version of coarse Alexander duality. As an application we provide a detailed proof of the following result, first stated by Kapovich-Kleiner. Given a $2$-complex formed by gluing $k$ halfplanes along their boundary lines and a coarse embedding into a contractible $3$-manifold, the complement consists of $k$ deep components that are arranged cyclically in a pattern called a Jordan cycle. We use the Jordan cycle as an invariant in proving the existence of a $3$-manifold group that is virtually Kleinian but not itself Kleinian.

math.GT↗
arXiv2024

Local connectedness of boundaries for relatively hyperbolic groups

Let $(Γ,\mathbb{P})$ be a relatively hyperbolic group pair that is relatively one ended. Then the Bowditch boundary of $(Γ,\mathbb{P})$ is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of $Γ$ and no restriction on the peripheral subgroups $P \in \mathbb{P}$.

math.GR↗
arXiv2024

Hyperbolic groups and local connectivity

The goal of this paper is to give an exposition of some results of Bestvina-Mess on local connectivity of the boundary of a one-ended word hyperbolic group. We also give elementary proofs that all hyperbolic groups are semistable at infinity and their boundaries are linearly connected in the one-ended case. Geoghegan first observed that semistability at infinity is a consequence of local connectivity using ideas from shape theory, and Bonk-Kleiner proved linear connectivity using analytical methods. The methods in this paper are closely based on the original ideas of Bestvina-Mess.

math.GR↗
arXiv2023

An introduction to semistability in geometric group theory

A finitely presented group is semistable at infinity if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable at infinity. This article provides a brief introduction to the notion of semistability at infinity in geometric group theory. We discuss techniques for proving semistability at infinity that involve either the topology of the boundary or the existence of certain hierarchies of splittings of the group. As an illustration of the second technique, we apply a combination theorem of Mihalik-Tschantz to prove semistability at infinity for groups that are hyperbolic relative to polycyclic subgroups using work of Mihalik-Swenson and Louder-Touikan.

math.GR↗
arXiv2022

Planar boundaries and parabolic subgroups

We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that if $(G,\mathcal{P})$ is a rigid relatively hyperbolic group pair whose boundary embeds in $S^2$, then the action on the boundary extends to a convergence group action on $S^2$. More generally, if the boundary is connected and planar with no cut points, we show that every element of $\mathcal{P}$ is virtually a surface group. This conclusion is consistent with the conjecture that such a group $G$ is virtually Kleinian. We give numerous examples to show the necessity of our assumptions.

math.GR↗
arXiv2021

On canonical splittings of relatively hyperbolic groups

A JSJ decomposition of a group is a splitting that allows one to classify all possible splittings of the group over a certain family of edge groups. Although JSJ decompositions are not unique in general, Guirardel--Levitt have constructed a canonical JSJ decomposition, the tree of cylinders, which classifies splittings of relatively hyperbolic groups over elementary subgroups. In this paper, we give a new topological construction of the Guirardel--Levitt tree of cylinders, and we show that this tree depends only on the homeomorphism type of the Bowditch boundary. Furthermore, the tree of cylinders admits a natural action by the group of homeomorphisms of the boundary. In particular, the quasi-isometry group of $(G,\mathbb{P})$ acts naturally on the tree of cylinders.

math.GR↗
arXiv2021

Cusped spaces and quasi-isometries of relatively hyperbolic groups

A group $Γ$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $Γ$ admits a cusp-uniform action on a proper $δ$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric, provided the spaces have "constant horospherical distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and by all negatively curved symmetric spaces. Consequently the Bowditch boundary admits a canonical quasisymmetric structure, which coincides with the "naturally occurring" quasisymmetric structure of the symmetric space when considering lattices in rank one symmetric spaces. We show that a group $Γ$ is a lattice in a negatively curved symmetric space $X$ if and only if a cusped space for $Γ$ is quasi-isometric to the symmetric space. We also prove an ideal triangle characterization of the $δ$--hyperbolic spaces with uniformly perfect boundary due to Meyer and Bourdon--Kleiner. An appendix concerns the equivalence of several definitions of conical limit point found in the literature.

math.GR↗
arXiv2020

Nonhyperbolic Coxeter groups with Menger curve boundary (with erratum)

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on $n$ vertices for $n\geq 5$. The construction depends on a slight extension of Sierpiński's theorem on embedding $1$--dimensional planar compacta into the Sierpiński carpet. See the appendix for a brief erratum.

math.GR↗
arXiv2019

Connectedness properties and splittings of groups with isolated flats

In this paper we study CAT(0) groups and their splittings as graphs of groups. For one-ended CAT(0) groups with isolated flats we prove a theorem characterizing exactly when the visual boundary is locally connected. This characterization depends on whether the group has a certain type of splitting over a virtually abelian subgroup. In the locally connected case, we describe the boundary as a tree of metric spaces in the sense of Świątkowski. A significant tool used in the proofs of the above results is a general convex splitting theorem for arbitrary CAT(0) groups. If a CAT(0) group splits as a graph of groups with convex edge groups, then the vertex groups are also CAT(0) groups.

math.GR↗
arXiv2018

Surface group amalgams that (don't) act on 3-manifolds

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of 3-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a 3-manifold. We prove that each such surface group amalgam is abstractly commensurable to a right-angled Coxeter group from a related family. In an appendix, we determine the quasi-isometry classes among these surface amalgams and their related right-angled Coxeter groups.

math.GT↗
arXiv2018

Distortion of surfaces in graph manifolds

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.

math.GR↗
arXiv2014

Finiteness properties of cubulated groups

We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let H_1,...,H_s be relatively quasiconvex codimension-1 subgroups of a group G that is hyperbolic relative to P_1,...,P_r. We prove that G acts relatively cocompactly on the associated dual CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact when G is hyperbolic. When P_1,...,P_r are abelian, we show that the dual CAT(0) cube complex C has a G-cocompact CAT(0) truncation.

math.GR↗
arXiv2010

Relative hyperbolicity and relative quasiconvexity for countable groups

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup.

math.GR↗
arXiv2009

Packing subgroups in relatively hyperbolic groups

We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of bounded packing, and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT(0) cube complexes. Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case.

math.GR↗
arXiv2008

Erratum on "Hadamard spaces with isolated flats"

The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats are extended to a more general setting in which the isolated subspaces are not necessarily flats. However, one step of that extension does not follow from the argument we used the isolated flats setting. We provide a new proof that fills this gap. In addition, we give a more detailed account of several other parts of Theorem A.0.1, which were sketched in math.GR/0411232.

math.GR↗