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.
Categories: math.GT math.DG