Representation zeta functions of split extensions of $SL_2^m(O)$

Abstract

We consider the representation growth of split extensions of $SL_2^m(O)$. We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of $SL_2^m(O)$ and the relative representation zeta function associated to the extension. We make use of our result by computing the zeta functions for two infinite families of split extensions of $SL_2^m(O)$ explicitly. Along the way, we compute the representation zeta functions of a large class of subgroups of $SL_2^m(O)$.