Equivariant torsion and base change

What is the true order of growth of torsion in the cohomology of an arithmetic group? Let [math] be a quaternion algebra over an imaginary quadratic field [math] . Let [math] be a cyclic Galois extension with [math] . We prove lower bounds for “the Lefschetz number of [math] acting on torsion cohomology” of certain Galois-stable arithmetic subgroups of [math] . For these same subgroups, we unconditionally prove a would-be-numerical consequence of the existence of a hypothetical base change map for torsion cohomology.