On angles in Teichm\"uller spaces

TL;DR

This paper explores the geometric properties of Teichmüller spaces, demonstrating the existence of geodesic triangles with arbitrary angles and side lengths, highlighting the complex angle behavior in infinite dimensions.

Contribution

It shows that in infinite dimensional Teichmüller spaces, geodesic triangles can have sides of equal length with arbitrary angles, including sums from 0 to 3π, revealing unusual geometric properties.

Findings

Existence of geodesic triangles with prescribed angles and side lengths

Angles in infinite-dimensional Teichmüller spaces can vary freely

Sum of angles in such triangles can range from 0 to 3π

Abstract

We discuss the existence of the angle between two curves in Teichm\"uller spaces and show that, in any infinite dimensional Teichm\"uller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length while its three angles are equal to any given three possibly different numbers from 0 to $\pi$. This implies that the sum of three angles of a geodesic triangle may be equal to any given number from 0 to $3\pi$ in an infinite dimensional Teichm\"uller space.