Modular Fibers And Illumination Problems

Details Modular Fibers And Illumination Problems Modular Fibers And Illumination Problems

2006-02-17

arxiv.org/abs/math/0602394v2

Mathematics

Geometric Topology

37 pages, 3 figures, submitted, 2/21/06 replacement contains one more figure

Scientific paper

For a Veech surface (x,\omega), we characterize subspaces of X^n, invariant under the diagonal action of the affine group of X. We prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X,\omega) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X,\omega) prelattice we prove the at most countableness of points non-illuminable from any x in X. Applying our results on invariant subspaces we prove the finiteness of these sets when (X,\omega) is Veech.

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