Veech surfaces and complete periodicity in genus two

Author:
Kariane Calta

Journal:
J. Amer. Math. Soc. **17** (2004), 871-908

MSC (1991):
Primary 37A99; Secondary 37E15, 37D40, 37D50

DOI:
https://doi.org/10.1090/S0894-0347-04-00461-8

Published electronically:
August 17, 2004

MathSciNet review:
2083470

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Abstract | References | Similar Articles | Additional Information

Abstract: We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair where is a Riemann surface and is an Abelian differential on . Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of .

A direction on a translation surface is completely periodic if any trajectory in the direction is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction . Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false.

In this paper, we use the -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.

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Additional Information

**Kariane Calta**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
kcalta@math.cornell.edu

DOI:
https://doi.org/10.1090/S0894-0347-04-00461-8

Received by editor(s):
January 13, 2003

Published electronically:
August 17, 2004

Article copyright:
© Copyright 2004
American Mathematical Society