Veech groups of Loch Ness monsters

TL;DR

This paper classifies the Veech groups of tame non-compact flat surfaces, showing that all countable subgroups of GL(2,R) avoiding small norm mappings can be realized as Veech groups of Loch Ness monster surfaces.

Contribution

It provides a complete classification of Veech groups for tame non-compact flat surfaces, including a realization result for certain subgroups of GL(2,R).

Findings

All countable subgroups of GL(2,R) avoiding small norm mappings appear as Veech groups.

Veech groups of tame flat surfaces are either countable or of three specific types.

The classification includes Loch Ness monster surfaces with prescribed Veech groups.

Abstract

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types.