Systoles on Heisenberg groups with Carnot-Carathéodory metrics (Q2759314)

The paper under review is a continuation of the author's previous one [Fundam. Prikl. Mat. 6, 401-432 (2000; Zbl 0983.22010)], the latter deals with the case of dimension 3, while the former is concerned with the higher dimensional case. Let \(H_{2n+1}\) be Heisenberg group of dimension \(2n+1\) and \(G=G(k_1,\cdots,k_n)\)be a uniform discrete subgroup. The quotient \(N^{2n+1}=H_{2n+1}/G\) is a compact nilmanifold. We endow \(N^{2n+1}\) with a Carnot-Carathéodory metric \(\rho_{\alpha}\) parametrized by a collection \((\alpha_1, \cdots, \alpha_n)\). The systole sys\((N^{2n+1}, \rho_{\alpha})\) is defined as the \(\rho_{\alpha}\)-length of a shortest closed loop in \(N^{2n+1}\) and the systolic constant is given by NEWLINE\[NEWLINE\sigma(N^{2n+1}, \rho_{\alpha})=\inf_{(k_1, \dots, k_n)}\frac{H^{2n+2}_{\rho_{\alpha}}(N^{2n+1}, \rho_{\alpha})} {(sys(N^{2n+1}, \rho+{\alpha}))^{2n+2}},NEWLINE\]NEWLINE where \(H^{2n+2}_{\rho_{\alpha}}\) is the \((2n+2)\)-dimensional Hausdorff measure generated by \(\rho_{\alpha}\). The author gives the following estimates for the systolic constant of the nilmanifold \(N^{2n+1}\) with Carnot-Carathéodory metric \(\rho_{\alpha}\): NEWLINE\[NEWLINE(\alpha_{\max}.\pi)^{-(n+1)}\mu_n<\sigma(N^{2n+1}, \rho_{\alpha})<(\alpha_{\min}.4/\sqrt{3})^{-(n+1)}\mu_n,NEWLINE\]NEWLINE where \(\alpha_{\min}=\min(\alpha_1, \cdots, \alpha_n)\) and \(\alpha_{\max}=\max(\alpha_1, \dots, \alpha_n)\), \(\mu_n\) is the proportion of the Hausdorff measure \(H^{2n+2}_{\rho_{\alpha}}\) with the \((2n+1)\)-dimensional Lebesgue measure \(L^{2n+1}\) of \(N^{2n+1}\).