Semistability of cubulated groups

TL;DR

This paper proves that all cubulated groups are semistable at infinity and introduces new cubulation techniques ensuring all halfspaces are one-ended and all quarterspaces are deep, advancing understanding of group boundaries.

Contribution

It establishes semistability for all cubulated groups and develops methods to modify cubulations to achieve specific geometric properties.

Findings

All cubulated groups are semistable at infinity.

Existence of cubulations with all halfspaces one-ended.

Existence of cubulations with all quarterspaces deep.

Abstract

We prove that all cubulated groups are semistable at infinity. In doing so we prove two further results about cubulations of groups. The first of these states that any one-ended cubulated group has a cubulation for which all halfspaces are one-ended. The second states that any cubulated group has a cubulation for which all quarterspaces are deep -- analogous to the fact that passing to the essential core of a given cubulation ensures that all halfspaces are deep.