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

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 $\mathrm{Cvol}(\Gamma \backslash G{)}^{\frac{1}{Q}}$, where $Q$ is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra $\mathfrak{g}=⨁V_i$. To prove this fact, the scalar product on $G$ introduced in the definition of Popp's volume plays a key role.

References in corpus (1)

• On the Hausdorff volume in sub-Riemannian geometry