Systoles of uniformly discrete lattices on higher Heisenberg groups with a Carnot-Caratheodory metric
by
Viktor Dontsov
Gubkin Oil and Gas University

Let (M, g) be a compact, smooth Riemannian n-manifold. Define a systole constant

\sigma(M):= inf g  \fracV0l(M, g)(sys(M, g))n,

where sys(M, g) is the smallest length of non-homotopy-trivial closed geodesics. Generalize this definition on a compact metric space (X, p) with an inner metric. Let m be the Hausdorff dimension of (X, p). Then the m- dimensional Hausdorff measure h_p^m =/= 0 generated by the metric p is defined on X.

Define the systole sys(X, p) of X as the smallest length of non-homotopy- trivial closed loops on X and let

\sigma(X)= inf p  \frachpm(X, p)(sys(X, p))m

be a systole constant for X.

Consider a 2n+1-dimension nilmanifold. X=H_2n+1/G, where H_2n+1 is a higher Heisenberg group, G subset H_2n+1 is a certain uniformly discrete subgroup (lattice). Let p be a left-invariant Carnot- Caretheodory metric (CC-metric) in H_2n+1.

It is know that the Hausdorff dimension of H_2n+1 equals 2n+2. Both h²ⁿ⁺² and the Lebesque measure L²ⁿ⁺¹ on H_2n+1 are Haar measures, hence h²ⁿ⁺²=\mu_n ·L²ⁿ⁺¹, where \mu_n is a universal multiplicative constant:

\frac\pi2n+1 <= \mun <= \frac\pi2n+1 ·\frac\pin-1n!\Thetan, where

\Thetan=\frac12 \pi ó õ 0  (\fracsintt)2n ·\fraccost ·sint-tcos2tt3dt

Theorem. For the systole constant \sigma(X) of X=H_2n+1/G with the CC-metric one has

\frac1\pin+1 ·\mun < \sigma(X) < (\frac\surd34)n+1\mun, where n > 1,

and \sigma(X)=\frac1116\pi²\mu₁, where \mu₁:

\frac\pi22 < \mu1 < \frac\pi22 / \Theta1,    \Theta1 \approx 0.826.

Date received: June 2, 1999