The maximum injectivity radius of hyperbolic orbifolds

TL;DR

This paper establishes a lower bound for the maximum embedded disk radius in hyperbolic orbifolds, identifying a unique extremal orbifold with specific cone point orders.

Contribution

It provides an explicit lower bound for the injectivity radius of hyperbolic orbifolds and characterizes the extremal case with three specific cone points.

Findings

The radius of a maximal embedded disk is at least 

Equality holds only for the orbifold with cone points of orders 2, 3, and 7

The explicit constant  is derived and proven optimal.

Abstract

For two-dimensional orientable hyperbolic orbifolds, we show that the radius of a maximal embedded disk is greater or equal to an explicit constant \rho_T, with equality if and only if the orbifold is a sphere with three cone points of order 2, 3 and 7.