  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.
