Symmetric Groups and Expander Graphs

TL;DR

This paper constructs explicit generating sets for symmetric and alternating groups that produce bounded degree expander graphs, solving a longstanding open problem with broad applications in random walks and combinatorics.

Contribution

It provides explicit constructions of generating sets for symmetric and alternating groups that yield families of bounded degree expander graphs, answering a major open question.

Findings

Constructed explicit generating sets for symmetric and alternating groups.

Produced families of bounded degree expander graphs for all n.

Confirmed the longstanding conjecture about the existence of such expanders.

Abstract

We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling and other areas.