Abstract

Veech Groups of Loch Ness Monsters

Highlights

  • For a compact flat surface S, the Veech group of S is the subgroup of SL(2, R) formed by the differentials of the orientation preserving affine homeomorphisms of S

  • In particular we prove that all countable subgroups of 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

  • Any subgroup G of GL+(2, R) satisfying assertion (i), (ii) or (iii) of Theorem 1.1 can be realized as a Veech group of a tame flat surface X which is a Loch Ness monster

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Summary

Introduction

For a compact flat surface S, the Veech group of S is the subgroup of SL(2, R) formed by the differentials of the orientation preserving affine homeomorphisms of S. Any subgroup G of GL+(2, R) satisfying assertion (i), (ii) or (iii) of Theorem 1.1 can be realized as a Veech group of a tame flat surface X which is a Loch Ness monster. We prove that any group satisfying assertion (ii) or (iii) can be realized as a Veech group of a tame flat surface which is a Loch Ness monster (Lemmas 3.7 and 3.8). We prove that any group satisfying assertion (i) of Theorem 1.1 can be realized as a Veech group of a tame flat surface which is a Loch Ness monster (Proposition 4.1). This construction is the main point of the article. We prove that if we assume in the hypothesis of Theorem 1.1 that G is countable, it satisfies assertion (i) (Lemma 4.15)

Preliminaries
Uncountable Veech groups
Countable Veech groups

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