The density of representation degrees

For a group G and a positive real number x, define dG(x) to be the number of integers less than x which are dimensions of irreducible complex representations of G. We study the asymptotics of dG(x) for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative ” for finitely generated linear groups G in characteristic zero, showing that either there exists α> 0 such that dG(x)> x α for all large x, or G is virtually abelian (in which case dG(x) is bounded)