Abstract：

A famous theorem of Gromov states that if a finitely generated group G has polynomial growth (i.e. there is a polynomial p such that the ball of radius n in some Cayley graph of G always contains at most p(n) vertices) then G has a nilpotent subgroup of finite index. Breuillard, Green and Tao proved a finitary refinement of this theorem, stating that if *some* ball of radius n contains at most eps n^d vertices then that ball contains a normal subgroup H such that G/H has a nilpotent subgroup with index at most f(d), where f is some non-explicit function. I will describe joint work with Romain Tessera in which we obtain explicit and even optimal bounds on both the index and the dimension of this nilpotent subgroup. I will also describe an application to random walks on groups.