Multiplicative subgroups of division algebras

(Joint work with Yoav Segev and Prasad and Rapinchuk). Inspired by the Kneser-Tits conjecture,
we study the multiplicative group $D^\times/N,$ where $D$ is a quaternion division algebra and $N$ is a normal
subgroup of $D^\times$ containing a noncentral element of order 2. Although this group is solvable in
positive characteristic $>2$, there is a counterexample in characteristic 0. The talk will focus on the techniques used to obtain these results.