Yiftach Barnea

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 dimH/ dimG (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 = SLd(Fp[[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(Fp[[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.

Abstract commensurators of profinite groups

In this paper we initiate a systematic study of the abstract com- mensurators of profinite groups. The abstract commensurator of a profinite group G is a group Comm(G) which depends only on the commensurability class of G. We study various properties of Comm(G); in particular, we find two natural ways to turn it into a topological group. We also use Comm(G) to study topological groups which contain G as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we con- struct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, like Pinks analogue of Mostows strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be re- formulated as structure theorems for the commensurators of certain profinite groups.

Index-subgroups of the Nottingham group

Abstract It is known that every countably based pro-p group embeds as a closed subgroup into the Nottingham group J = J ( F p ) . At the same time the subgroup structure of J remains rather mysterious. In this paper, we introduce the class of so-called index-subgroups of J and study their fundamental properties. Two main applications are given. (i) We prove that a certain infinite subclass of index-subgroups provides new examples of hereditarily just-infinite pro-p groups of finite width. (ii) We determine which Hausdorff dimensions occur for index-subgroups and subsequently show that the Hausdorff spectrum of J begins with a proper interval.

Graded subalgebras of affine Kac-Moody algebras

We determine the maximal graded subalgebras of affine Kac-Moody algebras. We also show that the maximal graded subalgebras of loop algebras are essentially loop algebras.

Lie algebras with few centralizer dimensions

Abstract It is known that a finite group with just two different sizes of conjugacy classes must be nilpotent and it has recently been shown that its nilpotence class is at most 3. In this paper we study the analogs of these results for Lie algebras and some related questions.

Subgroup growth in some pro- groups

For a group G let a n (G) be the number of subgroups of index n and let b n (G) be the number of normal subgroups of index n. We show that a p k(SL 1 2 (F p [[t]])) ≤ p k(k+5)/2 for p > 2. If A = F p [[t]] and p does not divide d or if A = Zp and p ¬= 2 or d ¬= 2, we show that for all k sufficiently large b p k(SL 1 d (Λ)) = b pk+d2-1 (SL 1 d (Λ)). On the other hand if Λ = F p [[t]] and p divides d, then b n (SL 1 d (Λ)) is not even bounded as a function of n.

Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract Let ℊ be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d (ℊ) = 2, where d(ℊ) is the number of generators of ℊ. Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G α} open subgroups which from a base for the topology at the identity and each G α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ≅ ℊ ⊗ tF p [t], where ℊ is a simple Lie algebra over F p , the field of p elements. We show that the lower rank of G is ≤ d (ℊ) + 1. We also show that if ℊ is simple classical of rank r and p > 7 or p 2r 2 − r, then the lower rank is actually 2.

Filtrations in semisimple lie algebras, I

This is the third in a series of papers. The first two, by Yiftach Barnea and this author, study the maximal bounded ℤ-filtrations of the finite-dimensional simple Lie algebras over the complex numbers. Those papers obtain a complete characterization for all but the five exceptional Lie algebras, namely the ones of type G 2, F 4, E 6, E 7 and E 8. Here, we fill in the missing step for the algebra G 2. The proof is computational and uses MAGMA, a computer algebra package, to handle the 7 × 7 matrices that occur.

On p-deficiency in groups

Recently, Schlage-Puchta proved super multiplicity of

Residual Properties of Free Pro-P Groups

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