Dimension and randomness in groups acting on rooted trees
By Miklós Abért, B'alint Vir'ag. 2002
71 citations recorded by the source corpus.
We explore the structure of the p-adic automorphism group of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of generated by three random elements are full-dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/42f374bf24097597d3a8610cb6c2ab2d6cf18111
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In Computational Theory and Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- GGS-groups: Order of congruence quotients and Hausdorff dimension (2011, background): “The determination of the Hausdorff dimension of closed subgroups of Γ has received special attention in the last few years (see [2,9,17,18]).”
- Hausdorff dimension in a family of self-similar groups (2006, background): “Abért and Virág showed in [1] that the closure of a group of p-adic tree automorphisms of the p-ary rooted tree generated by three randomly chosen automorphisms has Hausdorff dimension 1 with probability 1.”