Expansion in perfect groups

TL;DR

This paper characterizes when Cayley graphs of certain linear groups form expanders, linking this property to the perfection of the group's Zariski-closure's connected component, with implications for group expansion theory.

Contribution

It establishes a precise criterion connecting the expansion property of Cayley graphs to the perfection of the Zariski-closure's connected component of the group.

Findings

Cayley graphs form expanders for square-free q with large prime divisors if and only if the Zariski-closure's connected component is perfect.

Provides a characterization of expansion in terms of algebraic group properties.

Connects algebraic group structure to combinatorial expansion properties.

Abstract

Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an integer q, denote by Ga_q the subgroup of Ga consisting of the elements that project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q with respect to the generating set S form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Ga is perfect.