Right-angled Artin group boundaries

TL;DR

This paper demonstrates that right-angled Artin groups can have non-unique CAT(0) boundaries that are all arbitrarily connected, challenging previous assumptions about boundary connectivity in CAT(0) groups.

Contribution

It introduces examples of right-angled Artin groups with non-unique, arbitrarily connected CAT(0) boundaries and proves a combination theorem for certain amalgams of CAT(0) groups.

Findings

Existence of right-angled Artin groups with non-unique, arbitrarily connected CAT(0) boundaries

A new combination theorem for amalgams of CAT(0) groups with non-path connected boundaries

Application of the theorem to specific right-angled Artin groups

Abstract

In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of the boundaries are arbitrarily connected. We also prove a combination theorem for certain amalgams of CAT(0) groups to act on spaces with non-path connected visual boundaries. We apply this theorem to some right-angled Artin groups.