# Relative cubulations and groups with a 2-sphere boundary

## TL;DR

This paper introduces a new type of group action called relatively geometric action on CAT(0) cube complexes and uses it to characterize finite-volume Kleinian groups, extending previous results to the non-closed case.

## Contribution

It defines relatively geometric actions and applies them to characterize finite-volume Kleinian groups via cube complex actions, broadening the scope of geometric group theory.

## Key findings

- Introduction of relatively geometric actions on CAT(0) cube complexes
- Characterization of finite-volume Kleinian groups using these actions
- Extension of previous closed-case results to the non-closed case

## Abstract

We introduce a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of action on cube complexes, analogous to the results of Markovic and Ha\"issinsky in the closed case.