Non-existence of boundary maps for some hierarchically hyperbolic spaces

TL;DR

This paper investigates the existence of boundary maps in hierarchically hyperbolic spaces, showing non-existence in certain cases and establishing conditions for their existence in free subgroups of mapping class groups.

Contribution

It provides the first negative results on boundary maps for some hierarchically hyperbolic spaces and explores conditions for their existence in specific subgroups.

Findings

Boundary maps do not exist for some hierarchically hyperbolic spaces.

Boundary maps exist for free subgroups of mapping class groups.

Answers to open questions by Durham, Hagen, and Sisto.

Abstract

We provide negative answers to questions posed by Durham, Hagen, and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from right-angled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups.