Texas A&M University, Department of
Mathematics, 216 Milner Hall, 25th of October 2006, 3:00-3:50
Groups and Dynamics Seminar
Hausdorff dimension
in a family of self-similar groups
Zoran Šunić of Texas A&M
University
Finitely constrained groups are, in the theory of groups acting on
regular rooted trees, analogs of subshifts of finite type in symbolic
dynamics. In 2005 Grigorchuk showed that spherically transitive,
finitely constrained groups
are closed, self-similar, regular branch groups. We provide full
characterization of spherically transitive, finitely constrained groups
in terms of the
branching property. Namely, a group of tree automorphisms is
spherically transitive, finitely
constrained group if and only if it is the closure of a self-similar,
regular
branch group, branching over a congruence subgroup. We then provide a
formula for the Hausdorff dimension of finitely constrained groups and
show that infinite, finitely constrained groups of p-adic automorphisms
always have positive and rational Hausdorff dimension. We provide
concrete examples (not using random methods) of topologically finitely
generated self-similar groups of p-adic automorphisms whose Hausdorff
dimension is arbitrarily close to 1.