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 Γ∖𝐺 with Popp's volume. Namely, we show the existence of a positive constant C such that the systole of Γ∖𝐺 is less than Cvol(Γ∖𝐺)¹ᐟQ, where Q is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra 𝖌 = ⨁ 𝑉ᵢ. To prove this fact, the scalar product on 𝐺 introduced in the definition of Popp's volume plays a key role.