Perfect Matchings as IID Factors on Non-Amenable Groups

TL;DR

This paper demonstrates that in bipartite Cayley graphs of non-amenable groups, perfect matchings can be constructed as factors of independent uniform random variables, linking group properties with combinatorial structures.

Contribution

It introduces the novel result that perfect matchings in such graphs are factors of IID variables, extending the understanding of random processes on non-amenable groups.

Findings

Existence of IID factors forming perfect matchings in bipartite Cayley graphs of non-amenable groups

Improved Hoffman spectral bound on the independence number of finite graphs

Analysis of expansion properties of factors in these graphs

Abstract

We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs.