Compactifying a quotient with a cuspidal fundamental polygon #
A finite-sided convex polygon whose ideal vertices are cusp points supplies compact representatives for the complement of arbitrary cusp neighbourhoods, provided its carrier meets every orbit. The cusp data defining those neighbourhoods need not use the polygon's vertices as representatives, nor agree with the scalings chosen at those vertices.
If there are finitely many cusp orbits, these compact representatives imply compactness of
the cusp compactification. In particular, this applies to a cofinite group with such a polygon.
The polygon need only meet every orbit: neither uniqueness of representatives nor a
side pairing is needed for this compactness argument. Existence of such a polygon for an
arbitrary cofinite group, and hence compactness, is proved separately from its Dirichlet domain
(Subgroup.IsCofinite.compactSpace_compactifiedQuotient).
A side-paired polygon whose translates under a group Γ containing the side-pairing maps form a
locally finite family satisfies these hypotheses. Its translates cover ℍ, and each ideal vertex
is fixed by its cycle transformation, which is parabolic. Conversely, every cusp point of Γ is
Γ-equivalent to an ideal vertex, so Γ has finitely many cusp orbits and its compactified
quotient is compact. Disjointness of the translated interiors is not needed for any of this.
Main results #
ConvexPolygon.exists_cuspDatum_isCompact_carrier_diff_iUnion_horodisc: a cuspidal polygon minus horodiscs at its ideal vertices is compact.ConvexPolygon.exists_isCompact_cover_quotient_horodiscs: compact representatives outside an arbitrary family of cusp horodiscs.ConvexPolygon.compactSpace_compactifiedQuotient: compactness with finitely many cusp orbits.ConvexPolygon.isCompact_compl_iUnion_image_horodisc: compactness of the truncated coarse quotient.ConvexPolygon.SidePairing.isCuspPoint_of_vertex_eq_inr: the ideal vertices of a locally finite side-paired polygon are cusp points.ConvexPolygon.SidePairing.exists_vertex_eq_inr_smul_of_isCuspPoint: every cusp point is equivalent to an ideal vertex.ConvexPolygon.SidePairing.finite_cuspOrbit,ConvexPolygon.SidePairing.compactSpace_compactifiedQuotient: finitely many cusp orbits, and a compact compactified quotient.
References #
Katok, Fuchsian Groups, §4.2; Beardon, The Geometry of Discrete Groups, Chapter 9;
Diamond–Shurman, A First Course in Modular Forms, §2.4.
The geometric input is ConvexPolygon.isCompact_carrier_diff_iUnion, and the topological
input is Subgroup.CompactifiedQuotient.compactSpace_of_compact_truncations.
Truncating a cuspidal polygon at its ideal vertices. If every ideal vertex of a convex
polygon is a cusp point of Γ, then the ideal vertices carry cusp data based at them such that
the polygon minus horodiscs at these data, of arbitrary heights, is compact.
A convex polygon meeting every orbit and having only cuspidal ideal vertices gives a compact set of representatives outside any prescribed family of cusp horodiscs. The family may use arbitrary representatives and scalings, and its heights may be any real numbers.
Removing the images of a family of cusp horodiscs from the coarse quotient leaves a compact set when a cuspidal convex polygon meets every orbit. No finiteness assumption on the cusp set is needed for this truncated quotient.
A discrete group's cusp compactification is compact if it has finitely many cusp orbits and a finite-sided convex polygon with cuspidal ideal vertices meets every upper-half-plane orbit.
Ideal vertices of a locally finite side-paired polygon are cusp points. If a subgroup Γ
of PSL(2, ℝ) contains the cycle transformation at the ideal vertex vertex j and its translates
of P form a locally finite family, then vertex j is fixed by a parabolic element of Γ, namely
this cycle transformation.
Every cusp of a locally finite side-paired polygon is equivalent to an ideal vertex. If a
subgroup Γ of PSL(2, ℝ) contains every side-pairing map and its translates of P form a
locally finite family, then every cusp point of Γ is carried to an ideal vertex of P by an
element of Γ. Disjointness of the translated interiors is not needed.
A locally finite side-paired polygon has finitely many cusp orbits. If a subgroup Γ of
PSL(2, ℝ) contains every side-pairing map and its translates of P form a locally finite
family, then Γ has finitely many cusp orbits.
A locally finite side-paired polygon gives a compact compactified quotient. If a subgroup
Γ of PSL(2, ℝ) contains every side-pairing map and its translates of P form a locally finite
family, then the quotient Γ \ ℍ with its cusp orbits adjoined is compact.