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arXiv · 1605.03717

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

A. Muhammed Uludağ·Hakan Ayral
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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.

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A. Muhammed Uludağ, Hakan Ayral. 2016-05-12. On the involution of the real line induced by Dyer's outer automorphism of PGL(2,Z). https://arxiv.org/abs/1605.03717

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