A Fuchsian group is a subgroup 
of the projective
special linear group 
that is discrete in the inherited topology.
Its elements act on the upper
half-plane 
by Möbius transformations
 |
where 
and 
.
The matrices 
and 
define the same transformation.
These transformations are isometries
of the hyperbolic plane preserving its manifold
orientation (Voight 2021, p. 642).
Conjugation by the conformal mapping 
gives the equivalent action
on the unit disk (Voight 2021, p. 688). If 
is a torsion-free
group, its quotient space 
is a Riemann surface,
and the projection 
is its universal cover, with deck
transformation group 
. Every compact Riemann
surface of genus 
arises this way (Voight 2021, pp. 642-643).
