A vanishing theorem for the $p$-local homology of Coxeter groups

TL;DR

This paper proves a vanishing theorem for the p-local homology groups of Coxeter groups under specific conditions, extending known results for symmetric groups to a broader class of groups.

Contribution

It generalizes a vanishing result for symmetric groups to all Coxeter groups with certain prime-related conditions on generators.

Findings

p-local homology groups vanish for 1 ≤ k ≤ 2(p-2)

Generalizes Nakaoka's vanishing result for symmetric groups

Applicable to Coxeter groups with orders prime to p for generator products

Abstract

Given an odd prime number $p$ and a Coxeter group $W$ such that the order of the product $st$ is prime to $p$ for every Coxeter generators $s,t$ of $W$, we prove that the $p$-local homology groups $H_k(W,\mathbb{Z}_{(p)})$ vanish for $1\leq k\leq 2(p-2)$. This generalize a known vanishing result for symmetric groups due to Minoru Nakaoka.