Cancellation of Abelian groups of finite rank modulo elementary equivalence.
Cogrowth and spectral gap of generic groups
The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.
Conformally non-sperical 2-complexes.