Normal subgroups of the multiplicative group of a division algebra, and valuations

I will report on a joint work with Y.~Segev and G.M.~Seitz that states that for a finite dimensional division algebra $D$ over any field, all finite quotients of the multiplicative group $D^*$ are solvable. The purpose of the talk is to highlight the role of valuations in the argument. More precisely, I will discuss a congruence subgroup theorem for subgroups of finite index in $D^*$ and show how it quickly implies solvability of finite quotients. Some applications, such as a short proof of the Margulis-Platonov conjecture for ${\bf SL}_{1 , D}$ will be given.