Hausdorff dimension of closed subsets in profinite groups

Abstract

A countably based profinite group can be naturally seen as a metric space with respect to a given filtration, and thus, it has a well defined Hausdorff dimension function. Barnea and Shalev found a group theoretical expression for the Hausdorff dimension of the closed subgroups of a profinite group $G$, opening, in this way, a bunch of possibilities to explore. In this paper we generalize Barnea's and Shalev's result to arbitrary closed subsets of $G$.