The Octagonal PET I: Renormalization and Hyperbolic Symmetry

We introduce a family of polytope exchange transformations (PETs) acting on parallelotopes in ${\mathbb{R}}^{2n}$ for $n=1,2,\mathrm{3...}$. These PETs are constructed using a pair of lattices in ${\mathbb{R}}^{2n}$. The moduli space of these PETs is $G{L}_n(\mathbb{R})$. We study the case n=1 in detail. In this case, we show that the 2-dimensional family is completely renormalizable and that the $(2,4,\mathrm{\infty })$ 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.