On representations of direct products and the bounded generation property of branch groups

Abstract

We prove that the minimal representation dimension of a direct product G of non-abelian groups G_1,ldots ,G_n is bounded below by n+1 and thereby answer a question of Abért. If each G_i is moreover non-solvable, then this lower bound can be improved to be 2n. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups, we show that branch groups cannot be boundedly generated.