# Almost isometries between Teichm\"uller spaces

## TL;DR

This paper demonstrates that Teichm"uller spaces of surfaces with boundary, when equipped with certain metrics, are nearly isometric to spaces of punctured surfaces with corresponding metrics, revealing deep geometric connections.

## Contribution

It establishes almost isometries between different Teichm"uller spaces with boundary and punctured surfaces under specific metrics, providing new insights into their geometric relationships.

## Key findings

- Teichm"uller space with boundary and arc metric is almost isometric to punctured surface space with Thurston metric.
- Teichm"uller space with boundary and Teichm"uller metric is almost isometric to punctured surface space with the same metric.
- The results reveal a near-isometric correspondence between these moduli spaces under natural metrics.

## Abstract

We prove that the Teichm\"uller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichm\"uller metric) is almost isometric to the Teichm\"uller space of punctured surfaces equipped with the Thurston metric (resp. the Teichm\"uller metric).