[Paper Review] Boundary of the action of Thompson group F on dyadic numbers

Pavlo Mishchenko|arXiv (Cornell University)|Dec 9, 2015

Mathematical Dynamics and Fractals1 references6 citations

TL;DR

This paper proves that the Poisson boundary of the simple random walk on the Schreier graph of Thompson's group F acting on the dyadic rationals in [0,1] is non-trivial, providing a new proof that F lacks the Liouville property. Using electrical network theory and geodesic growth analysis, the author computes the geodesic growth function as $ \frac{1}{1-2z} $, confirming exponential growth and non-trivial harmonic functions.

ABSTRACT

We prove that the Poisson boundary of a simple random walk on the Schreier graph of action of $F$ on $\mathbb{D}$, where $\mathbb{D}$ is the set of dyadic numbers in $[0, 1]$, is non-trivial. This gives a new proof of the result of Kaimanovich: Thompson's group $F$ doesn't have Liouville property. In addition, we compute growth function of the Schreier graph of the action of $F$ on $\mathbb{D}$.

Motivation & Objective

• To establish the non-triviality of the Poisson boundary for the action of Thompson's group F on the dyadic rationals in [0,1].
• To provide an alternative proof that F does not have the Liouville property, i.e., admits non-constant bounded harmonic functions.
• To compute the geodesic growth function of the Schreier graph associated with the action $ F \curvearrowright \mathbb{D} $, where $ \mathbb{D} $ is the set of dyadic numbers in [0,1].
• To analyze the structure of cone types and derive recurrence relations for geodesic counts in the Schreier graph.

Proposed method

• Applies electrical network formalism to model the random walk on the Schreier graph, using resistance and Dirichlet energy to analyze transience and capacity.
• Uses a criterion based on the number of geodesics $ \texttt{gd}(x,n) $ from a fixed point to show that the capacity at the root is positive, implying transience.
• Applies Theorem 2.3 to show that if subtrees rooted at children of the root are transient, then the boundary of the random walk on the full tree is non-trivial.
• Derives recurrence relations for cone types (five in total) to model geodesic growth, using transition matrices and generating functions.
• Computes the extended complete geodesic growth function $ \tilde{L}(z) $ and the orbit growth function $ \widehat{L}(z) $ via matrix inversion of $ (I_5 - Az)^{-1} $.
• Uses the closed-form solution $ \overline{l}(z) = \left( \frac{1}{1-2z}, \frac{1}{1-z}, \frac{1}{1-3z+2z^2}, \frac{1}{1-3z+2z^2}, \frac{1}{1-2z} \right) $ to extract the geodesic growth function $ l(z) = \frac{1}{1-2z} $.

Experimental results

Research questions

• RQ1 Is the Poisson boundary of the simple random walk on the Schreier graph $ F \curvearrowright \mathbb{D} $ non-trivial?
• RQ2 Can the non-Liouville property of Thompson’s group F be re-proven using geometric and probabilistic methods on the Schreier graph?
• RQ3 What is the exact form of the geodesic growth function for the Schreier graph of $ F \curvearrowright \mathbb{D} $?
• RQ4 How do the cone types of vertices in the Schreier graph contribute to the growth and harmonic function structure?
• RQ5 What is the asymptotic growth rate of the number of points at distance $ n $ from the base point in the Schreier graph?

Key findings

• The Poisson boundary of the random walk on the Schreier graph $ F \curvearrowright \mathbb{D} $ is non-trivial, implying that Thompson’s group F does not have the Liouville property.
• The geodesic growth function of the Schreier graph is $ l(z) = \frac{1}{1-2z} $, indicating exponential growth with base 2.
• The number of vertices at graph distance $ n $ from the base point $ 1/2 $ is $ L_n = \frac{\varphi^{n+2} - \hat{\varphi}^{n+2}}{\sqrt{5}} $, where $ \varphi = \frac{1+\sqrt{5}}{2} $, showing Fibonacci-like growth.
• The orbit growth function $ \widehat{L}(z) $ is computed as $ \left( \frac{1+z}{1-z-z^2}, \frac{1}{1-z}, \frac{1+z}{1-2z+z^3}, \frac{1}{1-2z+z^3}, \frac{1}{1-z} \right) $, confirming the structure of cone types.
• The total number of vertices within distance $ n $ from the base point is $ |B(p,n)| = \frac{\varphi^{n+3} - \varphi^2}{\sqrt{5}(\varphi-1)} - \frac{\hat{\varphi}^{n+3} - \hat{\varphi}^2}{\sqrt{5}(\hat{\varphi}-1)} $, reflecting super-exponential accumulation.
• The recurrence relations for cone types and the matrix formulation $ (I_5 - Az)^{-1} \Lambda_0 $ yield a complete algebraic description of geodesic growth, enabling exact computation of growth functions.