Nilpotent Lie Algebras and Systolic Growth of Nilmanifolds

DOI10.1093/IMRN/RNX187zbMATH Open1439.22018arXiv1612.00818OpenAlexW3105728876MaRDI QIDQ5243359

Yves De Cornulier

Publication date: 18 November 2019

Published in: (Search for Journal in Brave)

Abstract: Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms of discrete cocompact subrings of the Lie algebra, making it more practical to estimate. After providing some general upper bounds, we develop methods to provide nontrivial lower bounds. We provide the first computations of the asymptotics of the systolic growth of nilpotent groups for which this is not equivalent to the volume growth. In particular, we provide an example for which the degree of growth is not an integer; it has dimension 7. Finally, we gather some open questions.

Full work available at URL: https://arxiv.org/abs/1612.00818

Mathematics Subject Classification ID

Subgroup theorems; subgroup growth (20E07) Geometric group theory (20F65) Differential geometry of homogeneous manifolds (53C30) Discrete subgroups of Lie groups (22E40) Nilpotent and solvable Lie groups (22E25) Asymptotic properties of groups (20F69) Solvable, nilpotent (super)algebras (17B30)

Related Items (3)

Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth ⋮ Nil Liep-algebras of slow growth ⋮ Nilmanifolds and associated Lie algebras over the integers.

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