Stabilizers of the finite vertices of a fundamental polygon #
Let σ be a side pairing of a convex hyperbolic polygon P, and let Γ ≤ PSL(2, ℝ) contain the
side-pairing maps and move the interior of P off itself: distinct translates of the interior
are disjoint. At a finite vertex z = vertex j, the tiles met by following the vertex cycle turn
around z through the successive interior angles. Two of these tiles whose sectors overlap must
be the same translate of P. This forces the angles of some number t of circuits of the
cycle to add up to exactly 2π, and the cycle transformation to have order t.
Every element of Γ fixing z then moves P to one of the tiles around z, so the stabilizer
of z in Γ is generated by the cycle transformation. Hence the stabilizer is finite cyclic
of order m and the angle sum of the cycle is 2π / m. This is the converse of the angle
condition at finite vertices in Poincaré's polygon theorem, and it reads off the elliptic points
of Γ on the boundary of P and their orders. Neither local finiteness of the translates nor
discreteness of Γ is assumed.
Main results #
ConvexPolygon.SidePairing.orderOf_cycleMap_mul_cycleAngleSum: the order of the cycle transformation at a finite vertex times its cycle angle sum is2π.ConvexPolygon.SidePairing.cycleAngleSum_eq_two_pi_div: equivalently, the cycle angle sum at a finite vertex is2πdivided by the finite order of the cycle transformation.ConvexPolygon.SidePairing.exists_partialCycleMap_eq_of_smul_vertex_eq: every transformation identifying two finite vertices occurs among the partial cycle products around the first.ConvexPolygon.SidePairing.smul_eq_self_iff_mem_zpowers_cycleMap: an element ofΓfixes the vertex exactly when it is a power of the cycle transformation.ConvexPolygon.SidePairing.map_stabilizer_eq_zpowers_cycleMap: the stabilizer of the vertex inΓis generated by the cycle transformation.ConvexPolygon.SidePairing.natCard_stabilizer_mul_cycleAngleSum: the order of the stabilizer times the cycle angle sum is2π.ConvexPolygon.SidePairing.cycleMap_eq_positiveStabilizerGenerator: the cycle transformation is the positive stabilizer generator, not its inverse.
References #
- Alan Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics 91, Springer, 1983, Chapter 9 (cycles of vertices of a fundamental polygon).
- Svetlana Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, Chapter 3 (elliptic cycles and their angle sums).
- Charles Walkden, Hyperbolic geometry (MATH32051 lecture notes, Manchester 2019), §17
(the angle sum of an elliptic cycle is
2π / m).
The angle condition at a finite vertex of a fundamental polygon. Let Γ contain the
side-pairing maps, with distinct translates of the interior of P disjoint. At a finite vertex,
the order of the cycle transformation times the cycle angle sum is 2π. In particular that
order is finite and the angle sum is 2π / m for a positive integer m.
At a finite vertex of a polygon with disjoint translated interiors, the cycle transformation has finite order.
At a finite vertex of a polygon with disjoint translated interiors, the cycle angle sum is
2π divided by the order of the cycle transformation.
The tiles containing a finite vertex are exactly the inverse partial cycle tiles before one full turn. This classifies all tiles at the vertex, without assuming that its preimage under the tile transformation is a vertex.
Every element identifying two finite vertices of a polygon with disjoint translated interiors
is a partial cycle product. The product occurs before the cycle has made one full turn, namely
before orderOf (cycleMap j) * cycleLength j steps.
The stabilizer of a finite vertex of a fundamental polygon. Let Γ contain the
side-pairing maps, with distinct translates of the interior of P disjoint. An element of Γ
fixes the finite vertex z = vertex j exactly when it is a power of the cycle transformation.
The stabilizer of a finite vertex of a fundamental polygon is generated by its cycle
transformation. Under the hypotheses of smul_eq_self_iff_mem_zpowers_cycleMap, the
stabilizer of the vertex in Γ, viewed in PSL(2, ℝ), is the cyclic group generated by the
cycle transformation.
A finite vertex has trivial stabilizer exactly when its cycle transformation is the identity. Thus the nontrivial finite cycles are precisely the elliptic vertex cycles.
The order of the stabilizer of a finite vertex equals the order of its cycle transformation, including the trivial cycle of order one.
The angle sum at a finite vertex of a fundamental polygon is 2π / m, where m is the
order of the stabilizer of the vertex. Under the hypotheses of
smul_eq_self_iff_mem_zpowers_cycleMap, the stabilizer of the vertex in Γ has order m with
m times the cycle angle sum equal to 2π.
The finite vertex-cycle transformation of a fundamental polygon is the positive stabilizer generator, not its inverse. Only disjointness of translated interiors is needed.