A classical lemma of Selberg says that every finitely generated linear group contains a subgroup of finite index which is torsion-free. We give quantitative estimates on the index in terms of the co-volume for arithmetic lattices. This can be used to deduce certain results for lattices with torsion which were known only for torsion-free lattices. 

We show that the space of traces of the free group is a Poulsen simplex, i.e., every trace is a pointwise limit of extremal traces. We prove that this fails for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras is a Poulsen simplex as well, answering a question of Musat and Rørdam.

We establish vanishing results for limits of characters in various discrete groups, most notably irreducible lattices in higher rank semisimple Lie groups. This is achieved by studying the geometry of the simplex of traces of discrete groups having Kazhdan's property (T) or its relative generalizations.

We show that the minimal number of generators of non-uniform higher rank lattices is sub-linear in their co-volume, partially answering a conjecture of Abert, Gelander and Nikolov. We prove a quantitative bound which in most cases is optimal. This implies that the first Betti number with coefficients in any ring grows sub-linearly. Arxiv version here.

Among other things, we show that lattices in an Amenable group are generated by at most C elements, where C depends only on the ambient group. This generalizes a result of Mostow from 1962. Arxiv version here.

We construct an efficient way to uniformly approximate the conformal map between two Lipschitz domains in the plane. Arxiv version here.