  The universal cover or the covering group of a hyperbolic Riemann surface $X$
is important but hard to express explicitly. It can be, however, detected by
the uniformisation and a suitable description of $X$. 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 $A$ in terms of five parameter pairs and give explicit
formulas about the hyperbolic structure and the complex structure of $A$.
Several degenerating cases are also treated.

Complex Variables