Quasi-semisimple profinite groups have polynomial representation growth precisely when their semisimple parts do, with equal growth degrees under bounded Lie rank, and constructions exist for any positive real growth degree with flexible composition factors.
Piccolo.On the representations of quasi-semisimple profinite groups, compactp-adic analytic groups, and Baumslag–Solitar groups
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Representation growth of quasi-semisimple profinite groups
Quasi-semisimple profinite groups have polynomial representation growth precisely when their semisimple parts do, with equal growth degrees under bounded Lie rank, and constructions exist for any positive real growth degree with flexible composition factors.