摘要

The universal cover or the covering group of a hyperbolic Riemann surface is important but hard to express explicitly. It can be, however, detected by the uniformization and a suitable description of . Beardon proposed five different ways to describe twice-punctured disks using fundamental domain, hyperbolic length, collar and extremal length in 2012. We parameterize a once-punctured annulus in terms of five parameter pairs and give explicit formulas about the hyperbolic structure and the complex structure of . Several degenerate cases are also treated.