[Paper Review] On a group associated to $z^2-1$

Laurent Bartholdi, Rostislav Grigorchuk|ArXiv.org|Mar 23, 2002

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper constructs the iterated monodromy group $G$ associated with the quadratic polynomial $f(z) = z^2 - 1$, showing it acts on a binary rooted tree and exhibits deep connections between group theory, dynamics, and geometry. The key result is that the Schreier graphs of $G$ converge in the Hausdorff metric to the Julia set of $f(z) = z^2 - 1$, establishing a direct link between group-theoretic objects 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 $f$.

Motivation & Objective

• To define and study the iterated monodromy group $G$ associated with the rational map $f(z) = z^2 - 1$ using covering space theory and tree automorphisms.
• To establish algebraic properties of $G$, including its weakly branch structure, torsion-freeness, and growth type.
• To explore the spectral properties of $G$'s regular representation and relate them to the dynamics of $f(z)$.
• To demonstrate the convergence of Schreier graphs of $G$ to the Julia set of $f(z) = z^2 - 1$ in the Hausdorff metric.
• To provide a complete presentation of $G$ and analyze its quotient structure and cohomological properties.

Proposed method

• Construct the iterated monodromy group $G = G(f)$ as a subgroup of automorphisms of a $2$-regular rooted tree via monodromy action on inverse iterates of a basepoint in the thrice-punctured sphere $\mathbb{C} \setminus \{0,1,\infty\}$.
• Use a wreath product presentation to describe $G$ via generators $a$ and $b$, where $a = \langle b,1 \rangle (1,2)$ and $b = \langle a,1 \rangle$, encoding self-similar action on subtrees.
• Analyze the group structure using recurrence relations and recursive constructions of Schreier graphs $\mathfrak{G}_n$ on $2^n$ vertices, with edges labeled by $a$ and $b$.
• Derive the spectrum of the regular representation $\pi$ on $L^2(\partial T)$ via a recursive polynomial system $Q_{n+1}(\lambda,\mu,\nu) = Q_n(F(\lambda,\mu,\nu))$, where $F(\lambda,\mu,\nu) = (\lambda^2 + 2\lambda\nu - 2\mu^2, \lambda\nu + 2\nu^2, -\mu^2)$.
• Prove that the spectrum of $\pi$ is the intersection of the line $\mu = \nu = -1/4$ with the closure of backward orbits of $\{Q_1 = 0\}$ under $F$.
• Establish the convergence of Schreier graphs $\mathfrak{G}_n$ to the Julia set $\mathfrak{J}$ of $z^2 - 1$ using planar embeddings and results from hyperbolic dynamics.

Experimental results

Research questions

• RQ1 How can the iterated monodromy group of $f(z) = z^2 - 1$ be explicitly constructed and described via self-similar tree automorphisms?
• RQ2 What are the algebraic properties of this group, such as its growth, torsion-freeness, and solvability of quotients?
• RQ3 How is the spectrum of the group's regular representation related to the dynamics of $f(z)$?
• RQ4 Can the Schreier graphs of the group be embedded in the complex plane so that they converge to the Julia set of $f(z)$?
• RQ5 What is the relationship between the group's cohomology, particularly its second homology, and its dynamical structure?

Key findings

• The group $G$ is weakly branch, meaning it contains a nontrivial normal subgroup that acts nontrivially on every level of the tree.
• The group $G$ is torsion-free, meaning no non-identity element has finite order.
• The group $G$ contains a free monoid of rank 2, implying exponential growth, but does not contain any non-abelian free group.
• Every proper quotient of $G$ is (free abelian)-by-(finite 2-group), and $G$ itself is not solvable.
• The group $G$ has a presentation with relations $[[a^p, b^p], b^p]$ and $[[b^p, a^{2p}], a^{2p}]$ for all $p$ a power of 2.
• The Schreier graphs $\mathfrak{G}_n$ converge in the Hausdorff metric to the Julia set $\mathfrak{J}$ of $z^2 - 1$, and are planar with a recursive structure based on $2^k$-gons and subtrees.