Affine Schottky Groups and Crooked Tilings

TL;DR

This paper explores the construction of proper affine deformations of Fuchsian Schottky groups using crooked planes, building on Drumm's and Margulis's foundational work from the 1980s and 1990s.

Contribution

It expounds Drumm's result specifically for Fuchsian Schottky groups, clarifying the construction of affine deformations via crooked planes.

Findings

Proper affine deformations exist for Fuchsian Schottky groups.

Crooked planes serve as a key tool in constructing these deformations.

The paper confirms the case when the group contains no parabolic elements.

Abstract

In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 1980's, answering a question raised by Milnor in 1977. This paper expounds Drumm's result, at least in the case of Fuchsian Schottky groups (that is, when the group contains no parabolic elements).