Quantum expanders and growth of group representations

TL;DR

This paper establishes exponential bounds on the growth of the number of irreducible group representations with bounded dimension for groups with spectral gap properties, revealing deep connections between quantum expanders and group representation theory.

Contribution

It provides the first explicit exponential bounds on the growth of irreducible representations in groups with spectral gaps, linking quantum expanders to representation growth.

Findings

Exponential bounds on the number of irreducible representations with dimension ≤ N.

Bounds depend on the spectral gap and the size of the generating set.

Results hold uniformly for large enough generating sets and all N ≥ 1.

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 $= N$.