[Paper Review] Cantor systems, piecewise translations and simple amenable groups

Kate Juschenko, Nicolas Monod|arXiv (Cornell University)|Apr 10, 2012

Advanced Operator Algebra Research2 references4 citations

TL;DR

This paper constructs the first known examples of finitely generated infinite simple groups that are amenable, resolving a long-standing open problem. By introducing a family of $L^2$-functions on the Bernoulli shift space and analyzing their invariance under piecewise translations of $\mathbb{Z}$, the authors prove the existence of an invariant mean on the group $W(\mathbb{Z})$ that concentrates on sets containing 0, which implies amenability of topological full groups of minimal Cantor systems and yields uncountably many non-isomorphic amenable simple groups.

ABSTRACT

We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space.

Motivation & Objective

• To resolve the open problem of whether there exist finitely generated infinite simple groups that are amenable.
• To prove that the topological full group of any minimal Cantor system is amenable.
• To construct a family of $L^2$-functions on the Bernoulli space $\{0,1\}^\mathbb{Z}$ with specific invariance properties under piecewise translations.
• To establish the existence of an invariant mean on $W(\mathbb{Z})$ that assigns full weight to sets containing 0, enabling the proof of amenability for full groups.

Proposed method

• Define a family of $L^2$-functions $f_n$ on the Bernoulli space $\{0,1\}^\mathbb{Z}$ using exponential decay weights based on distance from the origin.
• Use the inner product $\langle g(f_n), f_n \rangle / \|f_n\|^2$ to measure invariance of $f_n$ under the action of $g \in W(\mathbb{Z})$, showing convergence to 1 as $n \to \infty$.
• Introduce the function $F_n(g)$ to quantify signed deviations in translation behavior, proving $F_n(g) \to 0$ for all $g \in W(\mathbb{Z})$.
• Establish the 'ubiquitous pattern property' for subgroups of $W(\mathbb{Z})$, ensuring that local group actions repeat densely across $\mathbb{Z}$.
• Prove that stabilizers of finite symmetric differences with $\mathbb{N}$ are locally finite under this property, implying amenability.
• Apply the invariant mean construction to the topological full group $[[T]]$ of a minimal Cantor system via embedding into $W(\mathbb{Z})$, concluding amenability.

Experimental results

Research questions

• RQ1 Can there exist a finitely generated infinite simple group that is amenable, given that all previously known examples were either non-simple or non-amenable?
• RQ2 Does the topological full group of a minimal Cantor system admit an invariant mean, confirming Grigorchuk–Medynets's conjecture on amenability?
• RQ3 Can a family of $L^2$-functions on the Bernoulli shift be constructed such that their invariance under piecewise translations approaches 1 as $n \to \infty$?
• RQ4 Is there an invariant mean on the group $W(\mathbb{Z})$ of piecewise translations that assigns full measure to the collection of finite subsets containing 0?
• RQ5 Does the ubiquitous pattern property in $W(\mathbb{Z})$ imply that stabilizers of finite symmetric differences are locally finite, leading to amenability?

Key findings

• The topological full group of any minimal Cantor system is amenable, confirming the Grigorchuk–Medynets conjecture.
• There exist uncountably many non-isomorphic finitely generated infinite simple amenable groups, specifically $2^{\aleph_0}$ such groups.
• The $W(\mathbb{Z})$-action on the space of finite subsets of $\mathbb{Z}$ admits an invariant mean that assigns full weight to sets containing 0.
• For every $g \in W(\mathbb{Z})$, the inner product $\langle g(f_n), f_n \rangle / \|f_n\|^2 \to 1$ as $n \to \infty$, demonstrating asymptotic invariance of the function family.
• The function $F_n(g) \to 0$ for all $g \in W(\mathbb{Z})$, which is essential for proving the convergence of the inner product.
• The stabilizer of $E \triangle \mathbb{N}$ in any subgroup of $W(\mathbb{Z})$ with the ubiquitous pattern property is locally finite, which implies amenability of the full group.