Vanishing ranges for the mod $p$ cohomology of alternating subgroups of Coxeter groups

TL;DR

This paper establishes vanishing ranges for the mod p cohomology of alternating subgroups of finite p-free Coxeter groups, extending previous results for alternating groups and including twisted cohomology cases.

Contribution

It generalizes known vanishing results from alternating groups to broader classes of Coxeter groups, including infinite cases and twisted cohomology.

Findings

Vanishing ranges for mod p cohomology of finite p-free Coxeter groups.

Vanishing results for twisted cohomology with sign coefficients.

Partial results for certain infinite Coxeter groups.

Abstract

We obtain vanishing ranges for the mod $p$ cohomology of alternating subgroups of finite $p$-free Coxeter groups. Here a Coxeter group $W$ is $p$-free if the order of the product $st$ is prime to $p$ for every pair of Coxeter generators $s,t$ of $W$. Our result generalizes those for alternating groups formerly proved by Kleshchev-Nakano and Burichenko. As a byproduct, we obtain vanishing ranges for the twisted cohomology of finite $p$-free Coxeter groups with coefficients in the sign representations. In addition, a weak version of the main result is proved for a certain class of infinite Coxeter groups.