Ford and Dirichlet domains for cyclic subgroups of $PSL_2(\mathbb {C})$ acting on $\mathbb {H}^3_{\mathbb {R}}$ and $\partial \mathbb {H}^3_{\mathbb {R}}$

Abstract:

Let $\Gamma$ be a cyclic subgroup of $PSL_2({\mathbb C})$ generated by a loxodromic element. The Ford and Dirichlet fundamental domains for the action of $\Gamma$ on ${{\mathbb H}^3_{\mathbb R}}$ are the complements of configurations of half-balls centered on the plane at infinity ${\partial }{{\mathbb H}^3_{\mathbb R}}$. Jørgensen (On cyclic groups of Möbius transformations, Math. Scand. 33 (1973), 250–260) proved that the boundary of the intersection of the Ford fundamental domain with ${\partial }{{\mathbb H}^3_{\mathbb R}}$ always consists of either two, four, or six circular arcs and stated that an arbitrarily large number of hemispheres could contribute faces to the Ford domain in the interior of ${{\mathbb H}^3_{\mathbb R}}$. We give new proofs of Jørgensen’s results, prove analogous facts for Dirichlet domains and for Ford and Dirichlet domains in the interior of ${{\mathbb H}^3_{\mathbb R}}$, and give a complete decomposition of the parameter space by the combinatorial type of the corresponding fundamental domain.