# The density of representation degrees

### Martin W. Liebeck

Imperial College, London, UK### Aner Shalev

The Hebrew University of Jerusalem, Israel### Dan Segal

University of Oxford, United Kingdom

## 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).

## Cite this article

Martin W. Liebeck, Aner Shalev, Dan Segal, The density of representation degrees. J. Eur. Math. Soc. 14 (2012), no. 5, pp. 1519–1537

DOI 10.4171/JEMS/339