Nilpotent Lie algebras and systolic growth of nilmanifolds

TL;DR

This paper studies the systolic growth of nilpotent Lie groups, providing new methods to estimate and compute its asymptotics, including examples where growth differs from volume growth and has non-integer degrees.

Contribution

It introduces a practical approach to estimate systolic growth via discrete cocompact subrings and computes the first asymptotics for cases where it diverges from volume growth.

Findings

Systolic growth can grow faster than volume growth in nilpotent groups.

Provided the first asymptotic computations for non-volume-growth cases.

Discovered examples with non-integer growth degrees, such as dimension 7.

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.