A tree bijection for cusp-less planar hyperbolic surfaces

TL;DR

This paper extends a tree bijection for planar hyperbolic surfaces to include surfaces without cusps by utilizing half-tight cylinders and Busemann functions, enabling broader analysis of hyperbolic geometry.

Contribution

It introduces a generalized tree bijection applicable to all planar hyperbolic surfaces, removing the previous cusp requirement by employing half-tight cylinders and Busemann functions.

Findings

Extended the bijection to cusp-less surfaces.

Connected the decomposition into half-tight cylinders with the Busemann function.

Facilitated new computations of geometric properties on hyperbolic surfaces.

Abstract

Recently, a tree bijection has been found for planar hyperbolic surfaces, which allows for an easy computation of the Weil--Petersson volumes, and opens the path to get distance statistic on random hyperbolic surfaces and to find scaling limits when the number of boundaries becomes large. Crucially, this tree bijection requires the hyperbolic surface to have at least one cusp as origin, from which point distances are measured. In this paper we will extend this tree bijection, such that having a cusp is no longer required. We will first extend the bijection to half-tight cylinders. Since general planar hyperbolic surfaces can be naturally decomposed in two half-tight cylinders, this general case is also covered. In the half-tight cylinder the distances to the origin are replaced by the so-called Busemann function. This Busemann function is not well-defined on the surface, but it is on the cylinder cover.