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Séminaire de Géométrie

Siegel-Veech Constants of Cyclic Covers of Generic Translation Surfaces

David Aulicino

( (New York - Brooklyn College) )

Salle 2

le 23 février 2024 à 10:45

We consider generic translation surfaces of genus g>0g>0 with marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order dd. Given a translation surface, the number of cylinders with waist curve of length at most LL grows like L2L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L2L^2, the limit as LL goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points nn. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.