The density of representation degrees

Abstract

For a group $G$ and a positive real number $x$, define $d_G(x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$. We study the asymptotics of $d_G(x)$ 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 $\alpha >0$ such that $d_G(x)>x^{\alpha }$ for all large $x$, or $G$ is virtually abelian (in which case $d_G(x)$ is bounded).