Translation Surfaces arising from Right Regular Prisms

TL;DR

This paper investigates flat metrics from right regular n-prisms as translation surfaces, analyzing their unfoldings, orbit closures, and saddle connection counts, revealing new properties and explicit formulas.

Contribution

It introduces a novel analysis of right regular n-prisms as translation surfaces, showing their unfoldings are rarely lattice surfaces and establishing their orbit closures.

Findings

Unfoldings of right regular n-prisms are not lattice surfaces unless n=4.

These surfaces admit translation coverings to hyperelliptic surfaces.

Exact quadratic asymptotics for saddle connections and Siegel–Veech constants are derived.

Abstract

We study flat metrics arising from right regular $n$-prisms by viewing them as $n$-differentials and analyzing their associated unfoldings. We show that the unfolding of a right regular $n$-prism is never a lattice surface unless $n=4$, in contrast with the case of Platonic solids. Despite this, we prove that these surfaces admit translation coverings to hyperelliptic surfaces, allowing us to determine their $\mathrm{GL}(2,\mathbb{R})$-orbit closures using the classification of hyperelliptic components of strata. As a consequence, we obtain exact quadratic asymptotics for a certain average of the number of saddle connections on the base surfaces, their unfoldings, and the original prisms, including their Siegel--Veech constants. This provides a natural infinite family of non-lattice surfaces for which orbit closures and counting problems can be computed explicitly.