Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 058, 16 pages      arXiv:2201.00128      https://doi.org/10.3842/SIGMA.2022.058

Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume

Kenshiro Tashiro
Department of Mathematics, Tohoku University, Sendai Miyagi 980-8578, Japan

Received February 10, 2022, in final form July 28, 2022; Published online August 02, 2022

Abstract
In this paper, we give a systolic inequality for a quotient space of a Carnot group $\Gamma\backslash G$ with Popp's volume. Namely we show the existence of a positive constant $C$ such that the systole of $\Gamma\backslash G$ is less than ${\rm Cvol}(\Gamma\backslash G)^{\frac{1}{Q}}$, where $Q$ is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra $\mathfrak{g}=\bigoplus V_i$. To prove this fact, the scalar product on $G$ introduced in the definition of Popp's volume plays a key role.

Key words: sub-Riemannian geometry; Carnot groups; Popp's volume; systole.

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