Counting irreducible modules for profinite groups

This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property $\mathrm{PFP}_1$ is closed under extensions. In addition, we prove that profinite groups of type $\mathrm{FP}_1$ with UBERG are always finitely generated and we characterise UBERG in the class of pronilpotent groups. Using infinite products of finite groups, we construct several examples with unexpected properties: (1) a UBERG group which cannot be finitely generated, (2) a group of type $\mathrm{PFP}_\infty$ which is not UBERG and not finitely generated, and (3) a finitely generated group of type $\mathrm{PFP}_\infty$ with superexponential subgroup growth