On a group associated to $z^2-1$

TL;DR

This paper constructs a group acting on a binary rooted tree that models the monodromy of iterates of the quadratic polynomial z^2-1, revealing deep connections between algebra, geometry, and dynamics, including convergence to the Julia set.

Contribution

It introduces a novel group associated with z^2-1's dynamics and explores its algebraic properties and geometric connections, especially the convergence of Schreier graphs to the Julia set.

Findings

Schreier graphs converge to the Julia set

The group mimics monodromy actions of quadratic polynomial iterates

Connections established between group theory, geometry, and complex dynamics

Abstract

We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of $f(z)=z^2-1$ on associated coverings of the Riemann sphere. We then derive some algebraic properties of the group, and describe for that specific example the connection between group theory, geometry and dynamics. The most striking is probably that the quotient Cayley graphs of the group (aka ``Schreier graphs'') converge to the Julia set of $f$.