Lusztig's $a$ function for Coxeter groups of rank 3

TL;DR

This paper proves that Lusztig's $a$-function remains bounded for Coxeter groups of rank 3, providing new insights into the structure of these groups.

Contribution

It establishes the boundedness of Lusztig's $a$-function specifically for Coxeter groups of rank 3, a previously unresolved case.

Findings

Lusztig's $a$-function is bounded for rank 3 Coxeter groups

Provides a new understanding of the structure of rank 3 Coxeter groups

Advances the theory of Coxeter groups and their associated functions

Abstract

We show that Lusztig's $a$-function of a Coxeter group is bounded if the rank of the Coxeter group is 3.