Hausdorff dimension, pro groups, and KacMoody algebras
Authors:
Yiftach Barnea and Aner Shalev
Journal:
Trans. Amer. Math. Soc. 349 (1997), 50735091
MSC (1991):
Primary 28A78, 22C05; Secondary 20F40, 17B67
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 nonadic analytic groups , and study their spectrum. In particular we investigate the maximal Hausdorff dimension of nonopen 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 KacMoody 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:
http://dx.doi.org/10.1090/S0002994797019181
PII:
S 00029947(97)019181
Received by editor(s):
June 4, 1996
Additional Notes:
Supported by the United States – Israel BiNational Science Foundation, Grant No. 9200034/3
Article copyright:
© Copyright 1997
American Mathematical Society
