Quantum expanders and growth of group representations

Abstract

Let $\pi$ be a finite dimensional unitary representation of a group $G$ with a generating symmetric $n$-element set $S\subset G$. Fix \vp>0. Assume that the spectrum of $|S|^{-1}\sum_{s\in S} \pi(s) \otimes \overline{\pi(s)}$ is included in [-1, 1-\vp] (so there is a spectral gap \ge \vp). Let $r'_N(\pi)$ be the number of distinct irreducible representations of dimension $\le N$ that appear in $\pi$. Then let R_{n,\vp}'(N)=\sup r'_N(\pi) where the supremum runs over all $\pi$ with {n,\vp} fixed. We prove that there are positive constants \delta_\vp and c_\vp such that, for all sufficiently large integer $n$ (i.e. $n\ge n_0$ with $n_0$ depending on \vp) and for all $N\ge 1$, we have \exp{\delta_\vp nN^2} \le R'_{n,\vp}(N)\le \exp{c_\vp nN^2}. The same bounds hold if, in $r'_N(\pi)$, we count only the number of distinct irreducible representations of dimension exactly $= N$.Comment: Main addition: A remark due to Martin Kassabov showing that the numbers R(N) grow faster than polynomial. v3: Minor clarification