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

Abstract

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 p groups G G . We prove that if G G is p p -adic analytic and H ≤ c G H \le _c G is a closed subgroup, then the Hausdorff dimension of H H is dim ⁡ H / dim ⁡ G \dim H/\dim G (where the dimensions are of H H and G G as Lie groups). Letting the spectrum Spec ⁡ ( G ) \operatorname {Spec}(G) of G G denote the set of Hausdorff dimensions of closed subgroups of G G , it follows that the spectrum of p p -adic analytic groups is finite, and consists of rational numbers. We then consider some non- p p -adic analytic groups G G , and study their spectrum. In particular we investigate the maximal Hausdorff dimension of non-open subgroups of G G , and show that it is equal to 1 − 1 d + 1 1 - {1 \over {d+1}} in the case of G = S L d ( F p [ [ t ] ] ) G = SL_d(F_p[[t]]) (where p > 2 p > 2 ), and to 1 / 2 1/2 if G G is the so called Nottingham group (where p > 5 p >5 ). We also determine the spectrum of S L 2 ( F p [ [ t ] ] ) SL_2(F_p[[t]]) ( p > 2 p>2 ) completely, showing that it is equal to [ 0 , 2 / 3 ] ∪ { 1 } [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.