Hyperbolic models and stability recognition for Coxeter groups

Abstract

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.

arXiv commentv2 adds characterization of stable as undistorted and pure loxodromic, new intro, improved exposition, 69 pages, 7 figures