Veech groups of infinite-genus surfaces

Ramírez Maluendeas, Camilo; Valdez, Ferrán

doi:10.2140/agt.2017.17.529

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Veech groups of infinite-genus surfaces

Camilo Ramírez Maluendas and Ferrán Valdez

Algebraic & Geometric Topology 17 (2017) 529–560

DOI: 10.2140/agt.2017.17.529

Abstract

We show that every countable subgroup $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi> <mo class="MathClass-rel">&lt;</mo><msub><mrow><mo class="qopname"> GL</mo></mrow><mrow><mo class="MathClass-bin">+</mo></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn><mo class="MathClass-punc">,</mo> <mi>ℝ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ without contracting elements is the Veech group of a tame translation surface $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>$ of infinite genus for infinitely many different topological types of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>$ . Moreover, we prove that as long as every end has genus, there are no restrictions on the topological type of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>$ to realize all possible uncountable Veech groups.

Keywords

infinite type translation surface, Veech group

Mathematical Subject Classification 2010

Primary: 20F65, 53A99

References

Publication

Received: 26 April 2016

Revised: 1 June 2016

Accepted: 16 June 2016

Published: 26 January 2017

Authors

Camilo Ramírez Maluendas
Fundación Universitaria Konrad Lorenz Carrero 9 bis #62-43 Bogotá Colombia
http://docentes.konradlorenz.edu.co/2015/06/camilo-ramirez-maluendas.html
Ferrán Valdez
Center of Mathematical Sciences National Autonomous University of Mexico (UNAM) Campus Morelia C.P. 58190 Morelia Mexico
http://www.matmor.unam.mx/~ferran/

Volume 17, issue 1 (2017)