A geometric reduction theory for indefinite binary quadratic forms over $\mathbb{Z}[\lambda]$

TL;DR

This paper extends Gauss' classical reduction theory for indefinite binary quadratic forms from the integers to forms over $\\mathbb{Z}[\lambda]$, using symbolic dynamics on Hecke triangle surfaces to develop a geometric reduction framework.

Contribution

It introduces a new geometric reduction theory for indefinite binary quadratic forms over $\\mathbb{Z}[\lambda]$ associated with Hecke triangle groups, generalizing classical results.

Findings

Developed a reduction algorithm for forms over $\\mathbb{Z}[\lambda]$

Connected reduction theory to symbolic dynamics on Hecke triangle surfaces

Provided an algorithm with an upper runtime estimate for group membership decision

Abstract

Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely related to a certain symbolic dynamics for the geodesic flow on the modular surface, and hence can also be deduced geometrically. In this article, we use certain symbolic dynamics for the geodesic flow on Hecke triangle surfaces (also the non-arithmetic ones) to develop reduction theories for the indefinite binary quadratic forms associated to Hecke triangle groups. Moreover, we propose an algorithm to decide for any $g\in {\rm PSL}_2(\mathbb{R})$ whether or not $g$ is contained in the Hecke triangle group under consideration, and provide an upper estimate for its run time.