Bounding shears of spiralling triangulations on hyperbolic surfaces

We show that all hyperbolic surfaces admit an ideal triangulation with bounded shear parameters. This upper bound depends logarithmically on the topology of the surface.

💡 Research Summary

This paper establishes a significant uniform bound in hyperbolic geometry, analogous to the classical Bers constant but for a different type of decomposition. The main result (Theorem A) proves that for any hyperbolic surface X of genus g with n punctures (where 2g-2+n > 0), there exists an ideal triangulation whose shear parameters are bounded above by a constant that depends only logarithmically on the topology of the surface. Specifically, the “minimax shear” S(X) satisfies S(X) < 32·log(8π(2g-2+n)) + 23.

The research bridges the study of pants decompositions, where Bers constants bound the lengths of curves, and the theory of ideal triangulations described by shear coordinates. While previous work by Jiang provided bounds for triangulations with arcs ending in cusps, this paper handles the more general case where arcs are allowed to spiral infinitely around closed geodesics within the triangulation.

The proof strategy is elegant and constructive. It proceeds in two major steps. First, the author shows that any hyperbolic surface admits a “short” hexagon decomposition (Proposition 2.5). This is a decomposition of the surface into right-angled hexagons, obtained by cutting along a collection of disjoint simple closed geodesics Γ and a maximal set of orthogonal geodesic arcs A between them. Building on and extending work by Parlier, the author proves that such a decomposition exists where the lengths of curves in Γ and the lengths of arcs in A (or related truncated versions) are bounded by a constant times the logarithm of the area of the surface, which is 2π(2g-2+n).

Second, from this hexagon decomposition, an ideal triangulation T(Γ, A) is constructed. Each orthogonal arc a ∈ A is transformed into an ideal (bi-infinite) arc a∞ by making it spiral infinitely many times around the closed curve(s) at its endpoint(s). The collection of these spiraling arcs forms an ideal triangulation. A key observation is that the shear parameter along such a spiraling arc a∞ is essentially governed by the length of a finite truncation a∞^T of that arc, as the infinite spiraling part does not contribute to the shear due to an area argument. Consequently, the shear is bounded by the length of this truncated arc.

Since the original arc a from the hexagon decomposition was “short,” its truncated version a_t is also length-controlled. By relating the length of a∞^T to that of a_t, the author derives the universal logarithmic upper bound for the maximum shear parameter in the constructed triangulation, thus proving the main theorem.

The paper begins with a thorough background section, defining key concepts like Teichmüller space, ideal triangulations, shear parameters, collar neighborhoods, and cusp neighborhoods. It carefully explains how shear coordinates parameterize Teichmüller space and sets up the problem of bounding the minimax shear S(X). The final sections detail the proofs, combining geometric arguments about collars, area estimates, and the properties of hexagon decompositions. The result provides a new topological invariant for hyperbolic surfaces and enriches the understanding of their geometric structures through the lens of shear coordinates.