Realizing groups as Veech groups of dilation surfaces with infinite genus.

par Oscar Molina

mardi 1 avr. 2025, 11:15 → 12:15 Europe/Paris

1R2-207

A dilation surface is a topological surface equipped with a complex atlas whose transition maps, away from a discrete set of conical singularities, are dilations composed with translations.
Given a dilation surface M, an affine automorphism of M is an orientation-preserving homeomorphism from M to itself that leaves invariant the set of singularities and is locally a real affine map. The derivative of an affine automorphism is well defined up to positive rescaling and thus can be seen as an element of SL(2,R). The Veech group of M is the group consisting of all derivatives of affine automorphisms of M.
In this talk, we will study the Veech group of dilation surfaces whose fundamental group is not finitely generated and how it changes respect the Veech group of compact surfaces. In particular, we prove that  any infinite-countable subgroup of SL(2, R) can be realized, modulo isomorphism, as the Veech group of a finite (or infinite) area dilation surface homeomorphic to the Loch Ness Monster (the surface with infinite genus and only one end).
To prove this result we adapt ideas from the work of Artigiani, Randecker, Sadanand, Valdez and Weitze-Schmithüsen, who focused on translation surfaces.