The maximal injectivity radius of hyperbolic surfaces with geodesic boundary

TL;DR

This paper establishes precise upper bounds on the injectivity radii of finite-area hyperbolic surfaces with geodesic boundaries, extending previous results to include boundary components and discussing potential further extensions.

Contribution

It provides sharp upper bounds on injectivity radii for hyperbolic surfaces with boundary, generalizing prior work to include boundary components without fixing their lengths.

Findings

Sharp upper bounds for injectivity radii established

Bounds apply to all surfaces with fixed topology and boundary

Discussion on extending results via systole of loops

Abstract

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function.