The Octagonal PET I: Renormalization and Hyperbolic Symmetry

TL;DR

This paper introduces a family of polytope exchange transformations in even-dimensional spaces, studies their renormalization properties in the 2D case, and reveals hyperbolic symmetry actions on their moduli space, combining traditional and computer-assisted proofs.

Contribution

It constructs a new class of PETs using lattice pairs, analyzes their renormalization in 2D, and identifies hyperbolic symmetry groups acting on their moduli space.

Findings

The 2D family is completely renormalizable.

The hyperbolic (2,4,∞) reflection group acts on the moduli space.

The study combines classical mathematics with computer-assisted proofs.

Abstract

We introduce a family of polytope exchange transformations (PETs) acting on parallelotopes in $\R^{2n}$ for $n=1,2,3...$. These PETs are constructed using a pair of lattices in $\R^{2n}$. The moduli space of these PETs is $GL_n(\R)$. We study the case n=1 in detail. In this case, we show that the 2-dimensional family is completely renormalizable and that the $(2,4,\infty)$ hyperbolic reflection triangle group acts (by linear fractional transformations) as the renormalization group on the moduli space. These results have a number of geometric corollaries for the system. Most of the paper is traditional mathematics, but some part of the paper relies on a rigorous computer-assisted proof involving integer calculations.