Counting irreducible modules for profinite groups

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