Dimension and randomness in groups acting on rooted trees

Abért, Miklós; Virág, Bálint

doi:10.1090/S0894-0347-04-00467-9

Authors: Miklós Abért and Bálint Virág
Journal: J. Amer. Math. Soc. 18 (2005), 157-192
MSC (2000): Primary 20E08, 60J80, 37C20; Secondary 20F69, 20E18, 20B27, 28A78
DOI: https://doi.org/10.1090/S0894-0347-04-00467-9
Published electronically: September 2, 2004
MathSciNet review: 2114819
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Abstract: We explore the structure of the -adic automorphism group of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán.

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.

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