Stabilizers of ideal vertices in locally finite tessellations #
Let P be a convex hyperbolic polygon with an ideal vertex ξ, and let Γ ≤ PSL(2, ℝ) be a
subgroup whose translates of P form a locally finite family. Then every nonidentity element of
Γ fixing ξ is parabolic.
Such an element fixes a point of the ideal boundary, so it is not elliptic. If it were
hyperbolic, then after moving ξ to ∞ it would act as z ↦ μ * z + c with μ ≠ 1, and either
it or its inverse would contract towards its second fixed point x₀ ∈ ℝ. Near ∞, P is a
vertical strip, and the contracting powers carry it into thinner and thinner strips approaching the
vertical geodesic over x₀. So infinitely many distinct translates of P meet every neighbourhood
of x₀ + i, contradicting local finiteness.
Applied to the cycle transformations of a side-paired polygon, this gives the parabolic cycle
condition of Poincaré's polygon theorem at every ideal vertex of a locally finite fundamental
polygon (ConvexPolygon.SidePairing.isParabolic_cycleMap_of_locallyFinite).
Main result #
ConvexPolygon.isParabolic_of_smul_eq_self_of_locallyFinite: in a locally finite tessellation, a nonidentity element fixing an ideal vertex is parabolic.
References #
- Alan Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer, 1983, Chapter 9 (locally finite fundamental domains and their cycles).
- Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, Chapter 4 (fundamental regions and their vertices at infinity).
Nonidentity stabilizers of an ideal vertex are parabolic in a locally finite tessellation.
Let ξ be an ideal vertex of the convex polygon P, and let the translates of P by a subgroup
Γ of PSL(2, ℝ) form a locally finite family. Then every nonidentity element of Γ fixing ξ
is parabolic.