Cayley graph expanders and groups of finite width

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 equal to 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only two generators and four relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring M(3, F(2)). 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. We also conjecture that the group G has finite width 3 and finite average width 8/3. (C) 2011 Elsevier B.V. All rights reserved.