Cayley Graph Expanders and Groups of Finite Width

TL;DR

This paper introduces new infinite families of Cayley graph expanders with degree 4, based on finite quotients of a specific infinite group, and explores their algebraic properties and conjectures about the group's width.

Contribution

It constructs explicit Cayley graph expanders of minimal degree 4 using finite quotients of a specialized infinite group, and analyzes their algebraic and geometric properties.

Findings

Constructed two new families of Cayley graph expanders with degree 4.

Proved a 3-periodicity in the finite quotients of the group G.

Showed the pro-2 completion of G is infinite, non-analytic, and contains a free subgroup.

Abstract

We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only 2 generators and 4 relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring M(3,F2). We present explicit vector space bases for the finite abelian quotients of the lower exponent-2 groups of G by upper triangular subgroups and prove a particular 3-periodicity of these quotients. The pro-2 completion of the group G satisfies the Golod-Shafarevich inequality $|R| \geq (|X|^2)/4$, it is infinite, not p-adic analytic, contains a free nonabelian subgroup, but not a free pro-p group. We also conjecture that the group G has finite width 3 and finite average width 8/3.