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SUBGROUPS OF FRACTIONAL DIMENSION IN NILPOTENT OR SOLVABLE LIE GROUPS

Part of: Lie groups Classical measure theory

Published online by Cambridge University Press:  23 May 2013

Nicolas de Saxcé
Nicolas de Saxcé*
Affiliation:
Institute of Mathematics, Hebrew University, 91904 Jerusalem,Israel email saxce@math.huji.ac.il
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Abstract

We construct dense Borel measurable subgroups of Lie groups of intermediate Hausdorff dimension. In particular, we generalize the Erdős–Volkmann construction [Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math. 221 (1966), 203–208], showing that any nilpotent $\sigma $-compact Lie group $N$ admits dense Borel subgroups of arbitrary dimension between zero and $\dim N$. In algebraic groups defined over a finite extension of the rationals, using diophantine properties of algebraic numbers, we are also able to construct dense subgroups of arbitrary dimension, but the general case remains open. In particular, we raise the following question: does there exist a measurable proper subgroup of $ \mathbb{R} $ of positive Hausdorff dimension which is stable under multiplication by a transcendental number? Subgroups of nilpotent $p$-adic analytic groups are also discussed.

MSC classification

Secondary: 22E25: Nilpotent and solvable Lie groups 28A78: Hausdorff and packing measures
Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 497 - 511
Copyright
Copyright © University College London 2013 

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