## Dirichlet-Ford domains and double Dirichlet domains.(English)Zbl 1388.20068

Summary: We continue investigations started by G. S. Lakeland [Pac. J. Math. 255, No. 2, 417–437 (2012; Zbl 1257.20050)] on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic $$2$$- and $$3$$-space, or which have a Dirichlet domain with multiple centers. Such domains are called DF-domains and Double Dirichlet domains respectively. Making use of earlier obtained concrete formulas for the bisectors defining the Dirichlet domain of center $$i \in \mathbb H^2$$ or center $$j \in \mathbb H^3$$, we obtain a simple condition on the matrix entries of the side-pairing transformations of the fundamental domain of a Fuchsian or Kleinian group to be a DF-domain. Using the same methods, we also complement a result of Lakeland [loc. cit.] stating that a cofinite Fuchsian group has a DF domain (or a Dirichlet domain with multiple centers) if and only if it is an index $$2$$ subgroup of the discrete group G of reflections in a hyperbolic polygon.

### MSC:

 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 51M10 Hyperbolic and elliptic geometries (general) and generalizations 57M60 Group actions on manifolds and cell complexes in low dimensions

Zbl 1257.20050
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