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SUBGROUPS OF FRACTIONAL DIMENSION IN NILPOTENT OR SOLVABLE LIE GROUPS
Author(s) -
Saxcé Nicolas
Publication year - 2013
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579313000077
Subject(s) - mathematics , nilpotent , pure mathematics , hausdorff dimension , lie group , dimension (graph theory) , nilpotent group , algebraic number , transcendental number , combinatorics , discrete mathematics , mathematical analysis
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 σ ‐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 R of positive Hausdorff dimension which is stable under multiplication by a transcendental number? Subgroups of nilpotent p ‐adic analytic groups are also discussed.