A classification of right-angled Coxeter groups with no 3-flats and locally connected boundary

TL;DR

This paper classifies right-angled Coxeter groups with no 3-flats, showing that certain diagram properties determine whether all associated CAT(0) boundaries are locally connected, thus providing a complete classification.

Contribution

It establishes a classification of right-angled Coxeter groups with no 3-flats based on diagram separation properties and boundary local connectivity.

Findings

Absence of elementary separation implies locally connected boundary.

Previously known that separation property leads to non-locally connected boundary.

Complete classification of groups with no 3-flats and locally connected boundary.

Abstract

If $(W,S)$ is a right-angled Coxeter system and $W$ has no $\mathbb Z^3$ subgroups, then it is shown that the absence of an elementary separation property in the presentation diagram for $(W,S)$ implies all CAT(0) spaces acted on geometrically by $W$ have locally connected CAT(0) boundary. It was previously known that if the presentation diagram of a general right-angled Coxeter system satisfied the separation property then all CAT(0) spaces acted on geometrically by $W$ have non-locally connected boundary. In particular, this gives a complete classification of the right-angled Coxeter groups with no 3-flats and with locally connected boundary.