Bounding shears of spiralling triangulations on hyperbolic surfaces

TL;DR

This paper proves that every hyperbolic surface can be triangulated with shear parameters bounded by a logarithmic function of the surface's topology, providing a new geometric bound.

Contribution

It establishes a universal logarithmic upper bound on shear parameters for ideal triangulations of hyperbolic surfaces, advancing understanding of their geometric structure.

Findings

All hyperbolic surfaces admit ideal triangulations with bounded shear.

The shear bounds depend logarithmically on the surface's topology.

This result provides a new geometric constraint on hyperbolic surface triangulations.

Abstract

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.