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

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.