Symmetries of a two-dimensional continued fraction

TL;DR

This paper investigates the symmetry groups of two-dimensional continued fractions, particularly Klein polyhedra, identifying conditions for palindromic symmetries and relating them to algebraic properties of number fields.

Contribution

It provides a new criterion for the existence of palindromic symmetries in two-dimensional continued fractions, linking geometric symmetries to algebraic number theory.

Findings

Classification of symmetry types for two-dimensional continued fractions

A criterion for palindromic symmetry based on algebraic properties

Connection between geometric symmetries and number field units

Abstract

In this paper we describe the group of symmetries of a two-dimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: the Dirichlet-type ones, that correspond to the multiplication by units of the corresponding number field, and so called palindromic ones. The main result of the paper is a criterion for a two-dimensional continued fraction to have a palindromic symmetry. This criterion is analogous to the criterion for the period of a quadratic irrationality to be symmetric.