Dimension and randomness in groups acting on rooted trees

Abért, Miklós¹ ; Virág, Bálint²

¹Department of Mathematics, University of Chicago, 5734 University Ave., Chicago, Illinois 60637
²Department of Mathematics, University of Toronto, 100 St George St., Toronto, Ontario, Canada M5S 3G3

Journal of the American Mathematical Society, Tome 18 (2005) no. 1, pp. 157-192

Résumé

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.

Détail

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

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