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Christopher H. Cashen

Publications and source records attributed to Christopher H. Cashen.

At least 19 recordsLinked to original sources
arXiv2026

Hyperbolic models and stability recognition for Coxeter groups

Given a finite rank Coxeter system, we construct a hyperbolic space by coning off the wide parabolic subcomplexes of its Davis complex. The resulting space is ``stability recognizing'', in the sense that stable subspaces of the Davis complex are exactly the quasigeodesically connected subspaces whose images in the coned-off space are quasiisometrically embedded. Equivalently, the coned-off space is ``Morse recognizing'': a quasigeodesic in the Davis complex is Morse if and only if it is quasigeodesic in the coned-off space. Similar results are true for the mapping class group of a hyperbolic surface acting on its curve graph. We strengthen this analogy with additional results. First, the action of the Coxeter group on the coned-off space is acylindrical and gives the largest acylindrically hyperbolic structure for the Coxeter group. Second, the stable subgroups of a Coxeter group are characterized as the finitely generated subgroups that are undistorted and purely loxodromic with respect to the action on the coned-off space. The coned-off space, being constructed directly from the Coxeter structure, facilitates concrete characterizations of Morseness and stability for two important Coxeter-native families of objects: combinatorial geodesics and special subgroups.

math.GR↗
arXiv2026

Biggs tree groups

Biggs gave an explicit construction, using finite colored trees, of finite permutation groups whose Cayley graphs have valence \(C\) and girth tending to infinity as the radius \(R\) of the tree tends to infinity. We show that when the number of colors is at least 3, the group so presented contains the full alternating group on the vertices of the tree. This gives, for each \(C\geq 3\), an infinite family of pairs \((G_{C,R},S_{C,R})\) such that \(G_{C,R}\) is an alternating or symmetric group, \(S_{C,R}\) is a generating set of \(G_{C,R}\) of size \(C\) with an explicit permutation description of its generators, and such that the sequence of Cayley graphs \(\mathrm{Cay}(G_{C,R},S_{C,R})\) has constant valence \(C\) and girth tending to infinity as \(R\) tends to infinity.

math.GR↗
arXiv2026

Conformal dimension bounds, Pontryagin sphere boundaries, and algebraic fibering of right-angled Coxeter groups

We introduce a graph-theoretic condition, called $(n,m)$--branching, that ensures a combinatorial round tree with controlled branching parameters can be quasi-isometrically embedded in the Davis complex of the right-angled Coxeter group defined by the graph. This construction yields a lower bound on the conformal dimension of the boundary of such a hyperbolic group. We exhibit numerous families of graphs with this property, including many 1-dimensional spherical buildings. We prove an embedding result, showing that under mild hypotheses a flag-no-square graph embeds as an induced subgraph in a flag-no-square triangulation of a closed surface. We use this to embed our branching graphs into graphs presenting hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary. We conclude there are examples of such groups with conformal dimension tending to infinity, and hence, there are infinitely many quasi-isometry classes within this family. We use conformal dimension to show that recent work of Lafont--Minemyer--Sorcar--Stover--Wells can be upgraded to conclude that for every $n \geq 2$ there exist infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups that virtually algebraically fiber and have virtual cohomological dimension $n$.

math.GR↗
arXiv2025

RAAGedy right-angled Coxeter groups

We give criteria for deciding whether or not a triangle-free simple graph is the presentation graph of a right-angled Coxeter group that is quasiisometric to some right-angled Artin group, and, if so, producing a presentation graph for such a right-angled Artin group. We introduce two new graph modification operations, cloning and unfolding, to go along with an existing operation called link doubling. These operations change the presentation graph but not the quasiisometry type of the resulting group. We give criteria on the graph that imply it can be transformed by these operations into a graph that is recognizable as presenting a right-angled Coxeter group commensurable to a right-angled Artin group. In the converse direction we derive coarse geometric obstructions to being quasiisometric to a right-angled Artin group, first by specializing existing results from the literature to this setting, then by developing new approaches using configurations of maximal product regions. In all cases we give sufficient graphical conditions that imply these geometric obstructions. We implemented our criteria on a computer and applied them to an enumeration of small graphs. Our methods completely answer the motiving question when the graph has at most 10 vertices.

math.GR↗
arXiv2024

Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups

There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (Γ) and a subgraph (Λ) of its complement, checking some conditions, and, if satisfied, producing a graph (Δ) such that the right-angled Artin group with presentation graph (Δ) is a finite index subgroup of the right-angled Coxeter group with presentation graph (Γ). They do not tell us how to find (Λ), given (Γ). We show, in the 2--dimensional case, that the existence of such a (Λ) is connected to the graph property of satellite-dismantlabilty of (Γ), and we use this to give an algorithm for producing a suitable (Λ) or deciding that one does not exist.

math.GR↗
arXiv2024

Asymptotic cones of snowflake groups and the strong shortcut property

We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each of our groups has an asymptotic cone containing an isometrically embedded circle or, equivalently, has a Cayley graph that is not strongly shortcut. These are the first examples of groups whose asymptotic cones contain `metrically nontrivial' loops but no topologically nontrivial ones.

math.GR↗
arXiv2023

GAGTA 2023 Problem Session

The conference `Geometric and Asymptotic Group Theory with Applications (GAGTA) 2023: Groups and Dynamics' took place at the Erwin Schrödinger Institute on July 17-21. These are the problems that were proposed during the Problem Session: Residual finiteness growth, Geometric v. random walk boundaries, Stability, Cogrowth, Embedding left orderable groups, Schreier growth gap, $\ell^p$ models for hyperbolic groups, Characterizing hyperbolic groups

math.GR↗
arXiv2018

Morse subsets of CAT(0) spaces are strongly contracting

We prove that Morse subsets of CAT(0) spaces are strongly contracting. This generalizes and simplifies a result of Sultan, who proved it for Morse quasi-geodesics. Our proof goes through the recurrence characterization of Morse subsets.

math.GR↗
arXiv2018

Cogrowth for group actions with strongly contracting elements

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$, and let $δ_N$ and $δ_G$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric then $δ_N/δ_G>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk's original result on free groups with respect to a word metrics and a recent result of Jaerisch, Matsuzaki, and Yabuki on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

math.GR↗
arXiv2018

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ and not strongly contracting with respect to another. In the case of classical $C'(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical $Gr'(1/6)$ small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical $Gr'(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.

math.GR↗
arXiv2018

A Metrizable Topology on the Contracting Boundary of a Group

The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.

math.MG↗
arXiv2017

Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction

We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments.

math.MG↗
arXiv2016

Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology

We consider a `contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.

math.MG↗
arXiv2016

Quasi-isometries Between Groups with Two-Ended Splittings

We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these invariants reduce the problem of quasi-isometry classification of such groups to the problem of relative quasi-isometry classification of the factors of their JSJ decompositions. The first condition is that their JSJ decompositions have two-ended cylinder stabilizers. The second is that every factor in their JSJ decompositions is either `relatively rigid' or `hanging'. Hyperbolic groups always satisfy the first condition, and it is an open question whether they always satisfy the second. The same methods also produce invariants that reduce the problem of classification of one-ended hyperbolic groups up to homeomorphism of their Gromov boundaries to the problem of classification of the factors of their JSJ decompositions up to relative boundary homeomorphism type.

math.GR↗
arXiv2016

Splitting Line Patterns in Free Groups

We construct a boundary of a finite rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes. This provides a characterization of virtually geometric multiwords: they are the multiwords that are built from geometric pieces. In particular, a multiword is virtually geometric if and only if the relative boundary is planar.

math.GR↗