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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Howie, James / Khukhro, Evgenii I. / Kramer, Linus

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Hausdorff dimension of some groups acting on the binary tree

Olivier Siegenthaler1

1 Olivier Siegenthaler, EPFL SB IMB MAD, Station 8, CH-1015 Lausanne, Switzerland. E-mail:

Citation Information: Journal of Group Theory. Volume 11, Issue 4, Pages 555–567, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: 10.1515/JGT.2008.034, July 2008

Publication History

Accepted:
2006-11-11
Published Online:
2008-07-16

Abstract

Based on the work of Abercrombie [A. G. Abercrombie. Subgroups and subrings of profinite rings. Math. Proc. Cambridge Philos. Soc. 116 (1994), 209–222.], Barnea and Shalev [Y. Barnea and A. Shalev. Hausdorff dimension, pro-p groups, and Kac–Moody algebras. Trans. Amer. Math. Soc. 349 (1997), 5073–5091.] gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree 𝒯. Abért and Virág [M. Abért and B. Virág. Dimension and randomness in groups acting on rooted trees. J. Amer. Math. Soc. 18 (2005), 157–192.] showed that there exist finitely generated (but not necessarily level-transitive) subgroups of Aut 𝒯 of arbitrary dimension in [0, 1].

In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then give examples of 3-generated spinal groups which have transcendental Hausdorff dimension, and construct 2-generated groups whose Hausdorff dimension is 1.

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