Subgroup Structure, Fractal Dimension, and Kac-Moody Algebras

Abstract

This paper describes recent joint work with Barnea, and with Barnea and Zelmanov, where ideas from fractal geometry and Kac-Moody algebras are applied in studying the subgroup structure of profinite groups. We shall be interested in the spectrum of a finitely generated profinite group G, which is the set of Hausdorff dimensions of closed subgroups of G. This leads to questions about the subalgebra structure of affine Kac-Moody algebras. We determine the maximal graded subalgebras of affine Kac-Moody algebras, and derive applications to the spectrum of certain groups (such as matrix groups over local rings), whose subgroup structure is far from clear. We also examine the spectrum of p-adic analytic groups, and formulate several problems and conjectures.