A quantitative Neumann lemma for finitely generated groups

We study the coset covering function ℭ( r ) of an infinite, finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius r . We show that ℭ( r ) is of order at least r for all groups. Moreover, we show that ℭ( r ) is linear for a class of amenable gro...

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Bibliographic Details
Published in:Israel journal of mathematics 2024, Vol.262 (1), p.487-500
Main Authors: Gorokhovsky, Elia, Matte Bon, Nicolás, Tamuz, Omer
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Subgroups
Theoretical
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Summary:We study the coset covering function ℭ( r ) of an infinite, finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius r . We show that ℭ( r ) is of order at least r for all groups. Moreover, we show that ℭ( r ) is linear for a class of amenable groups including virtually nilpotent and polycyclic groups, and that it is exponential for property (T) groups.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-024-2617-x