Hausdorff dimension, pro- groups,

and Kac-Moody algebras

Authors:
Yiftach Barnea and Aner Shalev

Journal:
Trans. Amer. Math. Soc. **349** (1997), 5073-5091

MSC (1991):
Primary 28A78, 22C05; Secondary 20F40, 17B67

DOI:
https://doi.org/10.1090/S0002-9947-97-01918-1

MathSciNet review:
1422889

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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- groups . We prove that if is -adic analytic and is a closed subgroup, then the Hausdorff dimension of is (where the dimensions are of and as Lie groups). Letting the spectrum of denote the set of Hausdorff dimensions of closed subgroups of , it follows that the spectrum of -adic analytic groups is finite, and consists of rational numbers.

We then consider some non--adic analytic groups , and study their spectrum. In particular we investigate the maximal Hausdorff dimension of non-open subgroups of , and show that it is equal to in the case of (where ), and to if is the so called Nottingham group (where ). We also determine the spectrum of () completely, showing that it is equal to .

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.

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Additional Information

**Yiftach Barnea**

Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

Email:
yiftach@math.huji.ac.il

**Aner Shalev**

Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

Email:
shalev@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-97-01918-1

Received by editor(s):
June 4, 1996

Additional Notes:
Supported by the United States – Israel Bi-National Science Foundation, Grant No. 92-00034/3

Article copyright:
© Copyright 1997
American Mathematical Society