  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.
