Canonical bases for normal subgroups of finitely generated free groups with finite abelian factor groups.
We extend Rouquier’s categorification of the braid groups by complexes of Soergel bimodules to the virtual braid 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.