Polyhedral fundamental domains for discrete subgroups of PSL (2, R)

In his Ph.D. thesis [4], Thomas Fischer suggested how to construct a fundamental domain for the action of a discrete cocompact subgroup Γ of I+(H2), the group of orientation-preserving isometries of the hyperbolic plane H2, on this three-dimensional Liegroup I+(H2) by left translation. We recall Fischer’s construction and discuss properties of the tiling of I+(H2) by fundamental domains of Γ. In particular, its symmetries and their relation to the Seifert fibration of Γ\I+(H2) are considered.