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arXiv · 0809.1560

Cayley Graph Expanders and Groups of Finite Width

Norbert Peyerimhoff·Alina Vdovina
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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.

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BibTeXRIS

Norbert Peyerimhoff, Alina Vdovina. 2008-09-09. Cayley Graph Expanders and Groups of Finite Width. https://arxiv.org/abs/0809.1560

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