Systole Length in Hyperbolic $n$-Manifolds

TL;DR

This paper establishes a lower bound on the systole length of closed hyperbolic n-manifolds with a given triangulation, linking geometric properties to combinatorial complexity, and extends results to finite volume cases.

Contribution

It provides explicit bounds on systole length based on triangulation complexity and relates manifold diameter to the number of simplices, advancing understanding of hyperbolic manifold geometry.

Findings

Systole length is bounded below by a function of dimension and triangulation complexity.

Explicit bounds are given for the relation between core curve length and Margulis tube radius.

Results extend to finite volume hyperbolic manifolds with similar bounds.

Abstract

We show that the length $R$ of a systole of a closed hyperbolic $n$-manifold $(n \geq 3)$ admitting a triangulation by $t$ $n$-simplices can be bounded below by a function of $n$ and $t$, namely \[ R \geq \frac{1}{2^{(nt)^{O(n^4t)} }} .\] We do this by finding a relation between the number of $n$-simplices and the diameter of the manifold and by giving explicit bounds for a well known relation between the length of the core curve of a Margulis tube and its radius. We prove the same result for finite volume manifolds, with a similar but slightly more involved proof.