A Spectral Gap Theorem in $SU(d)$

TL;DR

This paper proves a spectral gap property for dense subgroups of the special unitary group $SU(d)$ generated by finitely many algebraic elements, extending previous results with a novel proof approach.

Contribution

It introduces a new proof method for establishing spectral gaps in $SU(d)$, generalizing earlier work from $SU(2)$ to higher dimensions.

Findings

Spectral gap established for dense subgroups of $SU(d)$

Method differs significantly from previous proofs in $SU(2)$

Results confirmed for groups generated by algebraic elements

Abstract

We establish the spectral gap property for dense subgroups of $SU(d)$ ($d\geq 2$), generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU(2).