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.
© de Gruyter 2008
