# On the connectedness of the boundary of hierarchically hyperbolic spaces

## TL;DR

This paper proves that under mild conditions, the boundary of a one-ended hierarchically hyperbolic space is connected, and explores the implications for group boundaries and their homeomorphisms.

## Contribution

It establishes the connectedness of boundaries in hierarchically hyperbolic spaces and groups, linking boundary connectedness to one-endedness, and characterizes boundary homeomorphisms for free products.

## Key findings

- Boundaries of one-ended hierarchically hyperbolic spaces are connected.
- Boundary connectedness is equivalent to one-endedness of the group.
- Homeomorphism of boundaries in free products depends on component boundaries.

## Abstract

We prove that, under a mild assumption, any metrizable compactification of a one-ended proper geodesic metric space is connected. As a consequence, we deduce that the boundary, introduced by Durham--Hagen--Sisto, of a one-ended hierarchically hyperbolic space is connected. Moreover, we prove that the connectedness of the boundary of a hierarchically hyperbolic group is equivalent to the one-endedness of the group. As an application, we show that if, for $n\geq 2$, $G_1=A_1\ast\dots\ast A_n$ and $G_2=B_1\ast\dots\ast B_n$ are free products of one-ended hierarchically hyperbolic groups, then the boundary of $G_1$ is homeomorphic to the boundary of $G_2$ if and only if the boundary of $A_i$ is homeomorphic to the boundary of $B_i$ for $1\leq i\leq n$.