Hyperbolic manifolds with a large number of systoles

TL;DR

This paper constructs sequences of hyperbolic n-manifolds with a number of systoles growing faster than volume, extending previous 2D and 3D results to higher dimensions with improved bounds.

Contribution

It generalizes known bounds on systoles in hyperbolic manifolds to all dimensions n ≥ 4, providing explicit growth rates relative to volume.

Findings

Number of systoles grows at least as volume^{1+1/(3n(n+1))−ε} for n≥4.

In dimension 3, the growth rate improves to volume^{4/3−ε}.

Extends previous 2D and 3D results to higher dimensions.

Abstract

In this article, for any $n\geq 4$ we construct a sequence of compact hyperbolic $n$-manifolds $\{M_i\}$ with number of systoles at least as $\mathrm{vol}(M_i)^{1+\frac{1}{3n(n+1)}-\epsilon}$ for any $\epsilon>0$. In dimension 3, the bound is improved to $\mathrm{vol}(M_i)^{\frac{4}{3}-\epsilon}$. These results generalize previous work of Schmutz for $n=2$, and D\'oria-Murillo for $n=3$ to higher dimensions.