thurston_veech

Thurston-Veech construction

A Thurston-Veech surface is a translation surface whose horizontal and vertical directions are parabolic, that is stabilized by a multitwist. Combinatorially, such surface decomposes into rectangles with horizontal and vertical sides.

We encode such surface by labelling the rectangles by \(\{1, 2, \ldots, n\}\). Then, the gluings is described by two permutations \(h\) and \(v\) that describe respectively horizontal gluings (going right) and vertical gluings (going up). The horizontal and vertical cylinders are then respectively the cycles of \(h\) and \(v\).

As shown by Thurston and Veech, to fix the flat structure of the surface one only has to specify the topological data of the twists, that is their multiplicities in each cylinder.

REFERENCES:

• Pascal Hubert, Erwan Lanneau “Veech groups without parabolic elements”

class flatsurf.geometry.thurston_veech.ThurstonVeech(hp, vp)

INPUT:

• hp - permutation describing the horizontal gluing of the rectangles

• vp - permutation describing the vertical gluing of the rectangles

EXAMPLES:

sage: from flatsurf.geometry.thurston_veech import ThurstonVeech
sage: TV = ThurstonVeech('(1,2)(3,4)', '(2,3)')
sage: TV
ThurstonVeech("(1,2)(3,4)", "(1)(2,3)(4)")
sage: TV.stratum()
H_2(1^2)

sage: S = TV([1,2], [3,1,1])
sage: S
Translation Surface in H_2(1^2) built from 4 rectangles
sage: S.base_ring()
Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?

sage: S = TV([1,1], [2,1,2])
sage: S.base_ring()
Rational Field

cylinder_intersection_matrix()

stratum()

stratum_component()