Cutting sequences, regular polygons, and the Veech group

Abstract

We describe the cutting sequences associated to geodesic flow on regular polygons, in terms of a combinatorial process called derivation. This work is an extension of some of the ideas and results in Smillie and Ulcigrai's recent paper, where the analysis was made for the regular octagon. It turns out that the main structural properties of the octagon generalize in a natural way.

Cutting sequences on translation surfaces

“…John Smillie and Corinna Ulcigrai gave a rule for the effect of the flip-shear on cutting sequences on the regular octagon surface, and on all regular 2n-gon surfaces [8], [9]. Using the same methods, we showed that the same rule holds for all double regular n-gon surfaces for odd n [3].…”

“…Recently, John Smillie and Corinna Ulcigrai studied the regular octagon surface, where the four pairs of parallel edges are identified [8], [9]. Subsequent work investigated the double pentagon surface, made by gluing the five pairs of parallel edges of a pair of regular pentagons [2], [3].…”

Cutting sequences on Bouw-Müller surfaces: an 𝓢-adic characterization

“…The full characterization of cutting sequences for the octagon, and more in general for regular polygon surfaces coming from the 2n-gons, was recently obtained by Smillie and the third author in the paper [38]; see also [37]. Shortly after the first author, Fuchs and Tabachnikov described in [12] the set of periodic cutting sequences in the regular pentagon, the first author showed in [13] that the techniques in Smillie and Ulcigrai's work [38] can be applied also to regular polygon surfaces with n odd. We now recall the characterization of cutting sequences for the regular octagon surface in [38], since it provides a model for our main result.…”

Cutting sequences on square-tiled surfaces

“…This question has been studied for a few special types of translation surfaces, which are surfaces equipped with a flat metric with cone-type singularities and trivial holonomy. In particular, cutting sequences have been described for flat tori [MH38] [Ser85]; the regular polygon surfaces with an even number of sides were studied by Smillie and Ulcigrai, [SU11]; Davis later extended the methods of Smillie and Ulcigrai to double n-gon surfaces with odd n [Dav13]; the L-shaped surface built from three squares was studied by Wu and Zhong, [WZ15]; and most recently the family of Bouw-Möller surfaces was studied by Davis, Pasquinelli, and Ulcigrai [DPU15]. Most of the previous results are obtained by considering the action of an element of the Veech group (group of affine symmetries) of the surface.…”