Mod $p$ and torsion homology growth in nonpositive curvature

Abstract

We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and L\"uck's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong.