Actions of right-angled Coxeter groups on the Croke–Kleiner spaces

Abstract

It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. In [4], Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces constructed in [4], then the local angles in those spaces all have to be π/2. We present a specific right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right angled Coxeter groups that can act geometrically on a given CAT(0) space are far from unique.This implies that the "right-angled" in the terminology "right-angled Coxeter group" turns out to be literal and is consistent with the "geometric" property of the group. Furthermore, given the result from [3] that if we fix the gluing angle of the Croke-Kleiner space at π/2 and change the side lengths of the tori, the resulting boundaries are not equivariantly homeomorphic to each other, we conclude the following: Corollary 1.2. There exists a right-angled Coxeter group that does not have unique equivariant visual boundary.

The large-scale geometry of right-angled Coxeter groups

“…In [Qin16a] and [Qin16b] Qing studied the actions of right-angled Coxeter groups on the Croke-Kleiner spaces from [CK00]. The group in the Croke-Kleiner example is the right-angled Artin group whose defining graph is the path of length four.…”

“…The corresponding Salvetti complex consists of three tori in a chain, so that the middle torus is glued to each of the two other tori along a simple closed curve, and the two simple closed curves in the middle torus intersect exactly once. (See [Qin16a,Fig 3] for a picture.) One then obtains an uncountable family of CAT(0) spaces by allowing the angle between the two simple closed curves to be anything in (0, π/2] and passing to the universal cover.…”

“…In [Qin16b], Qing shows that right-angled Coxeter groups are more "geometrically rigid" than right-angled Artin groups, in the following sense: if a right-angled Coxeter group acts geometrically on one of the Croke-Kleiner spaces above, then the angle must equal π/2. On the other hand, in [Qin16a] she considers the universal covers of the spaces obtained by keeping the angle between the two curves equal to π/2 but varying the lengths of the simple closed curves in the central torus (so that the tori are built out of rectangles instead of squares). These spaces are still CAT(0).…”

“…She shows: Theorem 4.5. [Qin16a] There exists a right-angled Coxeter group which acts geometrically on all the Croke-Kleiner spaces obtained by varying the lengths of the two simple closed curves in the central torus, but the boundaries of these spaces are not all equivariantly homeomorphic.…”

“…It is not known whether the boundaries of the spaces from [Qin16a] are homeomorphic, so the following question is still open: Question 4.6. Is there a right-angled Coxeter group with acts geometrically on two spaces whose boundaries are not homeomorphic?…”