The elliptic part of a Dirichlet domain

Let Γ ≤ PSL(2, ℝ) be discrete with finitely many elliptic orbits. A nonidentity element fixing a point w moves only a bounded set of points a bounded distance (Matrix.ProjectiveSpecialLinearGroup.isBounded_setOf_dist_smul_le_of_smul_eq_self). As there are finitely many elliptic orbits and each stabilizer is finite, the points moved a bounded distance by a nonidentity element of Γ fixing a point lie in the Γ-orbit of a bounded set. Their intersection with a Dirichlet domain is therefore bounded, because a point of the domain is the point of its orbit closest to the centre.

Main results

• Subgroup.isBounded_dirichletDomain_inter_setOf_dist_smul_le: for a discrete group with finitely many elliptic orbits, the points of a Dirichlet domain moved a bounded distance by a nonidentity element fixing a point form a bounded set.

References

• Alan Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer, 1983, §§9.4 and 10.1.

theorem Subgroup.isBounded_dirichletDomain_inter_setOf_dist_smul_le {Γ : Subgroup (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℝ)} [DiscreteTopology ↥Γ] (hΓ : Γ.ellipticOrbits.Finite) (p : UpperHalfPlane) (ε : ℝ) : Bornology.IsBounded (TauCeti.dirichletDomain (↥Γ) p ∩ {z : UpperHalfPlane | ∃ (g : ↥Γ), g ≠ 1 ∧ (∃ (w : UpperHalfPlane), g • w = w) ∧ dist (g • z) z ≤ ε})

The elliptic part of a Dirichlet domain is bounded. Let Γ ≤ PSL(2, ℝ) be discrete with finitely many elliptic orbits. For every centre p and every ε, the points of the Dirichlet domain of Γ centred at p moved a distance at most ε by some nonidentity element of Γ fixing a point of ℍ form a bounded set.