Arithmetic reflection groups and congruence subgroups

Abstract

This thesis investigates the geometric and topological constraints placed on the quotient space of a Fuchsian or Kleinian group by requiring that the group admits a fundamental domain which is simultaneously a Ford domain and a Dirichlet domain. In the case of Fuchsian groups, a direct correspondence with reflection groups is proved, and this result is used to first find explicitly the 23 non-cocompact arithmetic maximal hyperbolic reflection groups in the group of isometries of the hyperbolic plane, and subsequently to test whether these groups are all congruence. In the case of Kleinian groups, similar results are shown, and some examples of reflection groups are considered.