# Toy Teichm\"uller spaces of real dimension 2: the pentagon and the punctured triangle

## TL;DR

This paper investigates two 2-dimensional Teichmüller spaces associated with surfaces with boundary and marked points, revealing their unique geometric properties compared to closed surfaces, including polygonal exhaustion and linear divergence of geodesics.

## Contribution

It provides a detailed analysis of the geometry of the pentagon and punctured triangle Teichmüller spaces, highlighting their polygonal structure and geodesic behavior.

## Key findings

- Spaces are exhausted by regular convex polygons
- Geodesics diverge at most linearly
- Distinct geometric features from closed surface Teichmüller spaces

## Abstract

We study two $2$-dimensional Teichm\"uller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichm\"uller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed number of sides, and their geodesics diverge at most linearly.