# Hierarchies for Relatively Hyperbolic Virtually Special Groups

## TL;DR

This paper develops a new hierarchical framework for relatively hyperbolic virtually compact special groups, generalizing previous results and aiding in understanding their structure and quotient properties.

## Contribution

It constructs a virtual quasiconvex hierarchy for relatively hyperbolic virtually compact special groups, answering Wise's question and extending the Malnormal Special Quotient Theorem.

## Key findings

- Established a new hierarchy for relatively hyperbolic groups
- Generalized Wise's Malnormal Special Quotient Theorem
- Provided tools for analyzing group quotients with arbitrary peripherals

## Abstract

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually compact special groups. We use this hierarchy to prove a generalization of Wise's Malnormal Special Quotient Theorem for relatively hyperbolic virtually compact special groups with arbitrary peripheral subgroups.