The arithmetic of arithmetic Coxeter groups

TL;DR

This paper explores the connection between Conway's topograph, arithmetic Coxeter groups, and their applications to quadratic forms and Diophantine equations, extending the visualization method to broader algebraic structures.

Contribution

It introduces generalizations of Conway's topograph to various arithmetic Coxeter groups, linking geometric visualization with algebraic and number-theoretic concepts.

Findings

Established a framework connecting Coxeter groups and quadratic forms.

Extended the topograph visualization to new classes of arithmetic groups.

Potential applications to solving quadratic Diophantine equations.

Abstract

In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the "topograph," Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell's equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({\mathbb Z})$ and the Coxeter group of type $(3,\infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway's topograph, and generalizations to other arithmetic Coxeter groups. This includes a study of "arithmetic flags" and variants of binary quadratic forms.