On the involution of the real line induced by Dyer's outer automorphism of PGL(2,Z)

TL;DR

This paper investigates a special involution of the real line caused by Dyer's outer automorphism of PGL(2,Z), revealing its properties, functional equations, and effects on quadratic irrationals and the moduli space.

Contribution

It provides a detailed analysis of the involution's properties, its relation to the automorphism of an infinite tree, and conjectures about its action on algebraic numbers of degree three or higher.

Findings

The involution is discontinuous at rationals but satisfies functional equations.

It preserves real quadratic irrationals and commutes with Galois actions.

Conjecture: algebraic numbers of degree ≥3 are mapped to transcendental numbers.

Abstract

We study the involution of the real line induced by the outer automorphism of the extended modular group PGL(2,Z). This `modular' involution is discontinuous at rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationals mapping them in a non-obvious way to each other. It commutes with the Galois action on real quadratic irrationals. More generally, it preserves set-wise the orbits of the modular group, thereby inducing an involution of the moduli space of real rank-two lattices. We give a description of this involution as the boundary action of a certain automorphism of the infinite trivalent tree. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution.