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 <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 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 nonopen subgroups of G, and show that it is equal to 1 -1/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 SL₂ (Fv [[t]]) (p > 2) completely, showing that it is equal to [0,2/3] ∪ {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.