Divergence in right-angled Coxeter groups

Abstract

Abstract. Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex. IntroductionThe divergence of a pair of geodesics is a classical notion related to curvature. Roughly speaking, given a pair of geodesic rays emanating from a basepoint, their divergence measures, as a function of r, the length of a shortest "avoidant" path connecting their time-r points. A path is avoidant if it stays at least distance r away from the basepoint. In [15], Gersten used this idea to define a quasi-isometry invariant of spaces, also called divergence. We recall the definitions of both notions of divergence in Section 2.The divergence of every pair of geodesics in Euclidean space is a linear function, and it follows from Gersten's definition that any group quasi-isometric to Euclidean space has linear divergence. In a δ-hyperbolic space, any pair of non-asymptotic rays diverges exponentially; thus the divergence of any hyperbolic group is exponential. In symmetric spaces of non-compact type, the divergence is either linear or exponential, and Gromov suggested in [16] the same should be true in CAT(0) spaces.Divergence has been investigated for many important groups and spaces, and contrary to Gromov's expectation, quadratic divergence is common. Gersten first exhibited quadratic divergence for certain CAT(0) spaces in [15]. He then proved in [14] that the divergence of the fundamental group of a closed geometric 3-manifold is either linear, quadratic or exponential, and characterised the (geometric) ones with quadratic divergence as the fundamental groups of graph manifolds. showed that all graph manifold groups have quadratic divergence. More recently, Duchin-Rafi [13] established that the divergence of Teichmüller space and the mapping class group is quadratic (for mapping class groups this was also obtained by Behrstock in [5]). Druţu-Mozes-Sapir [12] have conjectured that the divergence of lattices in higher rank semisimple Lie groups is always linear, and proved this conjecture in some cases. Abrams et al [1] and independently Behrstock-Charney [2] have shown that if A Γ is the right-angled Artin group associated to a graph Γ, the group A Γ has either linear or quadratic divergence, and its divergence is linear if and only if Γ is (the 1-skeleton of) a join.In this work we study the divergence of 2-dimensional right-angled Coxeter groups. Our first main result is Theorem 1.1 below, which characterises such groups with linear and quadratic divergence in terms of their defining graphs. This result can be seen as a step in the quasi-isometry classification of (right-angled) Coxeter groups, about which very little is known.We note that by [10], every right-angled Artin group is a finite index subgroup of, and therefore quasi-isometric to, a right-angled Coxeter group. However, in contrast to the setting of right-angled Artin gr...

Top-dimensional quasiflats in CAT(0) cube complexes

“…Another interesting fact in the case d = 1 is that one can tell whether Γ admits a non-trivial join decomposition by looking at the diameter of G 1 (Γ). This basically follows from the argument in [DT12]. See Theorem 5.30 for a precise statement.…”

On strongly quasiconvex subgroups

“…Let γ 1 be the subpath of γ connecting x and α d (r). Then the length of γ 1 is at least 1 2 d(d+1) r d by the proof of Proposition 5.3 in [DT15a]. Therefore, the length of γ is also at least 1 2 d(d+1) r d .…”

The relative hyperbolicity and manifold structure of certain right-angled Coxeter groups

“…In geometric group theory, groups acting on CAT(0) cube complexes are fundamental objects and right-angled Coxeter groups provide a rich source such groups. The coarse geometry of right-angled Coxeter groups has been studied by Caprace [Cap09,Cap15], Dani-Thomas [DT15,DT], Dani-Stark-Thomas [DST], Behrstock-Hagen-Sisto [BHS17], Levcovitz [Lev18] and others. In this paper, we will study the boundary of relatively hyperbolic right-angled Coxeter groups and the geometry of right-angled Coxeter groups with planar nerves.…”