Betti numbers and injectivity radii

TL;DR

This paper establishes lower bounds on the maximal injectivity radius of certain hyperbolic 3-manifolds based on their Betti numbers and topological features, improving understanding of their geometric properties.

Contribution

It provides new lower bounds for the maximal injectivity radius in hyperbolic 3-manifolds with specific Betti number conditions and topological restrictions, extending prior results.

Findings

Maximal injectivity radius exceeds 0.32798 for manifolds with Betti number 2 and no fibroid surface.

Compared to previous bounds, this improves the known lower bounds under certain topological conditions.

The proof combines existing results with techniques developed by the authors in the 1990s.

Abstract

We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds 0.32798. For comparison, Andrew Przeworski showed, with no topological restrictions, that the maximal injectivity radius exceeds arcsinh(1/4) = 0.247..., while the authors showed that if M has first Betti number at least 3 then the maximal injectivity exceeds log(3)/2 = 0.549.... The proof combines a result due to Przeworski with techniques developed by the authors in the 1990s.