Infinite translation surfaces with infinitely generated Veech groups

Veech Groups of Loch Ness Monsters

“…We refer the reader to [6,10] for more details on Veech groups of compact flat surfaces, and to [3,4,9,2] for explicit examples of Veech groups of tame flat surfaces which are Loch Ness monsters.…”

A series of coverings of the regular n-gon

“…is the lattice surface Γ l 2,0 defined in [HS10] and the limit of our series is their lattice surface Γ ∞ 2,0 . For n = 6, considerations similar to Proposition 2.6. in [Sch04] show that every affine map f on Y (respecting Σ(Y ) = p −1 (Σ(X 6 )) ) descends via a translation covering p : Y → X 6 to an affine mapf on the twice punctured torus X 6 with p • f =f • p, i.e.…”

“…It is build from two infinite polygons, the convex hull of the points (n, n 2 ) and the convex hull of the points (n, −n 2 ). Another example is the infinite staircase origami Z ∞ 2,0 , calculated in [HS10].…”

Generalized Staircases: Recurrence and Symmetry

“…Removing the restriction that the surface is compact gives a flexible setup and many phenomena, absent in the compact case, may be observed. For example, in the recent paper [8], Hubert and Schmithüsen made the surprising discovery that there are infinite square tiled surfaces whose Veech group is infinitely generated.…”

“…The examples studied in [9,8] are Z-covers of compact translation surfaces. Although this class is much smaller than the general case, it already displays many surprising features.…”