# Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3

## TL;DR

This paper proves Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3, providing a detailed description of their a-function and Kazhdan-Lusztig cells, advancing understanding in geometric group theory.

## Contribution

The paper establishes the validity of Lusztig's conjectures P1-P15 specifically for rank 3 hyperbolic Coxeter groups, a case previously unresolved.

## Key findings

- Confirmed Lusztig's conjectures P1-P15 for these groups
- Provided explicit descriptions of the a-function and Kazhdan-Lusztig cells
- Enhanced understanding of the structure of hyperbolic Coxeter groups

## Abstract

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description of the a-function and Kazhdan-Lusztig cells for these Coxeter groups.