Fractal dimensions and profinite groups

Let T T be a finitely branching rooted tree such that any node has at least two successors. The path space [ T ] [T] is an ultrametric space: for distinct paths f , g f,g let d ( f , g ) = 1 / | T n | d(f,g)= 1/|T_n| , where T n T_n denotes the n n -th level of the tree, and n n is largest such that f ( n ) = g ( n ) f(n)= g(n) . Let S S be a subtree of T T without leaves that is level-wise uniformly branching, in the sense that the number of successors of a node only depends on its level. We show that the Hausdorff and lower box dimensions coincide for [ S ] [S] , and the packing and upper box dimensions also coincide. We give geometric proofs, as well as proofs based on the point-to-set principles. We use the first result to reprove a theorem of Barnea and Shalev [Trans. Amer. Math. Soc. 349 (1997), pp. 5073–5091] on the Hausdorff dimension of closed subgroups of a profinite group G G , referring only on the geometric structure of the closed subgroup in the canonical path space given by an inverse system for G G . We obtain an analogous theorem for the packing dimension.