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Bounded Generation and Linear Groups

A group Γ is called boundedly generated (BG) if it is the set-theoretic product of finitely many cyclic subgroups. We show that a BG group has only abelian by finite images in positive characteristic representations. We use this to reprove and generalize Rapinchuk's theorem by showing that a BG group with the FAb property has only finitely many irreducible representations in any given dimension over any field. We also give a structure theorem for the profinite completion G of such a group Γ. On the other hand, we exhibit boundedly generated profinite FAb groups which do not satisfy this structure theorem.