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Dimension and randomness in groups acting on rooted trees

Author(s): Miklós Abért; Bálint Virág
Journal: J. Amer. Math. Soc. 18 (2005), 157-192.
MSC (2000): Primary 20E08, 60J80, 37C20; Secondary 20F69, 20E18, 20B27, 28A78
Posted: September 2, 2004
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Abstract: We explore the structure of the $p$-adic automorphism group $Y$ 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 $W$ 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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Additional Information:

Miklós Abért
Affiliation: Department of Mathematics, University of Chicago, 5734 University Ave., Chicago, Illinois 60637
Email: abert@math.uchicago.edu

Bálint Virág
Affiliation: Department of Mathematics, University of Toronto, 100 St George St., Toronto, Ontario, Canada M5S 3G3
Email: balint@math.toronto.edu

DOI: 10.1090/S0894-0347-04-00467-9
PII: S 0894-0347(04)00467-9
Keywords: Groups acting on rooted trees, Galton-Watson trees, Hausdorff dimension, pro $p$-groups, generic subgroups, symmetric $p$-group
Received by editor(s): February 16, 2003
Posted: September 2, 2004
Additional Notes: The first author's research was partially supported by OTKA grant T38059 and NSF grant \#DMS-0401006.
The second author's research was partially supported by NSF grant \#DMS-0206781 and the Canada Research Chair program.
Copyright of article: Copyright 2004, M. Ab\'ert and B. Vir\'ag

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