Systoles of nilmanifolds with the Carnot-Caratheodory metrics (Q1405607)

Let \((X, \rho)\) be a metric space of Hausdorff dimension \(m\) with an inner metric \(\rho\). The systolic constant of \(X\) is the infimum over all inner metrics on \(X\) of the quotient of the \(m\)-dimensional Hausdorff volume of \(X\) and the \(m\)-th power of the smallest length of homotopy nontrivial closed loops on \(X\). Such constant is homotopy invariant for Riemannian manifolds and it is very difficult to compute. In the present paper the author considers the class of nilmanifolds with the Carnot-Caratheodory metrics and obtains some estimates for the systolic constants of these metric spaces.