Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras

Every finitely generated profinite group can be given the structure of a metric space, and as such it has a well defined Hausdorff dimension function. In this paper we study Hausdorff dimension of closed subgroups of finitely generated pro-$p$ groups $G$. We prove that if $G$ is $p$-adic analytic and $H \le _c G$ is a closed subgroup, then the Hausdorff dimension of $H$ is $\dim H/\dim G$ (where the dimensions are of $H$ and $G$ as Lie groups). Letting the spectrum $% \operatorname {Spec}(G)$ of $G$ denote the set of Hausdorff dimensions of closed subgroups of $G$, it follows that the spectrum of $p$-adic analytic groups is finite, and consists of rational numbers.

We then consider some non-$p$-adic analytic groups $G$, and study their spectrum. In particular we investigate the maximal Hausdorff dimension of non-open subgroups of $G$, and show that it is equal to $1 - {1 \over {d+1}}$ in the case of $G = SL_d(F_p[[t]])$ (where $p > 2$), and to $1/2$ if $G$ is the so called Nottingham group (where $p >5$). We also determine the spectrum of $SL_2(F_p[[t]])$ ($p>2$) completely, showing that it is equal to $[0,2/3] \cup \{ 1 \}$.

Some of the proofs rely on the description of maximal graded subalgebras of Kac-Moody algebras, recently obtained by the authors in joint work with E. I. Zelmanov.