# Teichm\"uller theory and collapse of flat manifolds

## TL;DR

This paper offers an algebraic framework for understanding the Teichmüller and moduli spaces of flat metrics on manifolds and orbifolds, analyzing their boundaries and collapse phenomena, especially in three dimensions.

## Contribution

It introduces an algebraic description of these spaces and classifies collapsed limits of flat 3-manifolds, revealing new insights into their geometric degeneration.

## Key findings

- Boundary of moduli space consists of flat orbifolds
- Every flat orbifold can be obtained by collapsing flat manifolds
- Classified collapsed limits of flat 3-manifolds

## Abstract

We provide an algebraic description of the Teichm\"uller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by collapsing closed flat manifolds, and the collapsed limits of closed flat 3-manifolds are classified.