[Paper Review] Coning-off CAT(0) cube complexes

Anthony Genevois|arXiv (Cornell University)|Mar 21, 2016

Geometric and Algebraic Topology14 references17 citations

TL;DR

This paper introduces a coning-off construction for CAT(0) cube complexes over combinatorially convex subcomplexes to study their hyperbolicity. It establishes that killing uniformly thick flat rectangles via cone-off yields hyperbolic spaces, leading to new proofs of relative hyperbolicity for right-angled Coxeter groups and acylindrical hyperbolicity for $C^'(1/4)$-$T(4)$ small cancellation quotients of free products.

ABSTRACT

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.

Motivation & Objective

• To characterize when CAT(0) cube complexes are hyperbolic by analyzing flat rectangles and hyperplane grids.
• To develop a coning-off method that transforms non-hyperbolic cube complexes into hyperbolic spaces by eliminating thick flat rectangles.
• To reprove and refine the relative hyperbolicity of right-angled Coxeter groups using cubical techniques.
• To establish acylindrical hyperbolicity for $C^'(1/4)$-$T(4)$ small cancellation quotients of free products via cone-off and hyperbolicity criteria.
• To generalize weak acylindrical actions to higher-dimensional cube complexes, improving on prior results in the hyperbolic setting.

Proposed method

• Define two types of cone-offs: one via edge additions over subcomplexes, and the standard cone-off with new vertices for each subcomplex.
• Introduce the concept of $L$-thin and $L$-thick flat rectangles and relate their uniform thinness to global hyperbolicity of the complex.
• Establish a criterion for the usual cone-off to be fine, based on local finiteness and bounded hyperplane intersections with subcomplexes.
• Use the cone-off construction to prove that hyperbolicity is achieved when thick flat rectangles are uniformly bounded in the cone-off.
• Apply the hyperbolicity criterion to contact graphs and hyperplane stabilizers to derive weak relative hyperbolicity results.
• Leverage the equivalence between weak acylindrical and acylindrical actions in finite-dimensional cube complexes to prove acylindrical hyperbolicity of quotients.

Experimental results

Research questions

• RQ1 Under what conditions is a CAT(0) cube complex hyperbolic, and how do flat rectangles and hyperplane grids relate to this?
• RQ2 Can the cone-off construction over combinatorially convex subcomplexes produce hyperbolic spaces, and what conditions ensure this?
• RQ3 How can the cone-off method be used to reprove the strong relative hyperbolicity of right-angled Coxeter groups?
• RQ4 What conditions on small cancellation quotients of free products ensure acylindrical hyperbolicity?
• RQ5 Is there a generalization of weak acylindrical actions to higher-dimensional CAT(0) cube complexes, and how does this relate to acylindrical hyperbolicity?

Key findings

• A CAT(0) cube complex is hyperbolic if and only if its flat rectangles are uniformly thin, or equivalently, if its hyperplane grids are uniformly thin.
• The cone-off of a CAT(0) cube complex over a family of combinatorially convex subcomplexes is hyperbolic if the thick flat rectangles are uniformly bounded in the cone-off.
• The cone-off over non-$n$-combinatorially contracting hyperplanes yields a hyperbolic graph, improving Hagen’s result on weak relative hyperbolicity.
• For right-angled Coxeter groups, the group is weakly hyperbolic relative to the stabilizers of non-contracting hyperplanes, and this extends to specific collections of subgroups such as $C( ilde{ ho}_1 ilde{ ho}_2)$ or $ ext{star}(u)$ for vertices in induced squares.
• The usual cone-off of a uniformly locally finite CAT(0) cube complex is fine if and only if the subcomplexes are locally finite and hyperplane intersections are uniformly bounded.
• For $C^'(1/4)$-$T(4)$ small cancellation quotients of free products with uniformly bounded free product lengths, the quotient group admits an acylindrical universal action, and such groups are acylindrically hyperbolic when $k \geq 5$ in the example $K_{k,I}$.