The arc metric on Teichm\"uller spaces of surfaces of infinite type with boundary

TL;DR

This paper introduces an arc metric on the Teichmüller space of infinite-type hyperbolic surfaces with boundary, based on the Basmajian identity, and explores its properties and examples.

Contribution

It defines a new asymmetric metric on Teichmüller spaces of infinite-type surfaces using the Basmajian identity and provides examples satisfying the necessary geometric conditions.

Findings

Defined an arc metric on Teichmüller space of infinite-type surfaces.

Constructed examples of hyperbolic surfaces satisfying the geometric condition.

Explored the relation between Shiga's condition and the geometric condition.

Abstract

Let $X_{0}$ be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric (which is called arc metric) on the quasiconformal Teichm\"uller space $\mathcal{T}(X_{0})$ provided that $X_{0}$ satisfies a geometric condition. Furthermore, we construct several examples of hyperbolic surfaces of infinite type satisfying the geometric condition and discuss the relation between the Shiga's condition and the geometric condition.