Dirichlet–Ford domains and arithmetic reflection groups

Grant S. Lakeland

Vol. 255 (2012), No. 2, 417–437

DOI: 10.2140/pjm.2012.255.417

Abstract

We show that a Fuchsian group, acting on the upper half-plane model for ℍ², admits a Ford domain which is also a Dirichlet domain, for some center, if and only if it is an index 2 subgroup of a reflection group. This is used to exhibit an example of a maximal arithmetic hyperbolic reflection group which is not congruence. Analogous results, and counterexamples, are given in the case of Kleinian groups.

Keywords

fundamental domain, arithmetic reflection groups, congruence

Mathematical Subject Classification 2010

Primary: 20H10

Secondary: 19B37

Milestones

Received: 8 March 2011

Accepted: 16 June 2011

Published: 10 April 2012