Counting Formulae for Square-tiled Surfaces in Genus Two

Shrestha, Sunrose T.

Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 83-123.

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4\pi$ each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for enumeration of square-tiled surfaces in these four diagrams, completing the detailed count for genus two. The formulae are in terms of various well-studied arithmetic functions, enabling us to give asymptotics for each diagram. Interestingly, two of the four cylinder diagrams occur with asymptotic density 1/4, but the other diagrams occur with different (and irrational) densities.

Classification : 00X99
Mots clés : Square-tiled surface, translation surface, counting, primitive

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