Polynomial representation growth and alternating quotients.

Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the

number of isomorphism classes of irreducible $n$-dimensional complex

characters of $\Gamma$. Representation growth is the study of the

behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of

representation growth.

We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is

bounded by a polynomial in $n$. I will discuss a question posed by

Brent Everitt: can a group with polynomial representation growth have

the alternating group $A_n$ as a quotient for infinitely many $n$?