[Paper Review] Free subgroups in groups acting on rooted trees

Volodymyr Nekrashevych|ArXiv.org|Feb 18, 2008

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper establishes a structural dichotomy for groups acting on rooted trees: if a group contains a free non-abelian subgroup, it must either act freely at a boundary point or have a free subgroup acting faithfully on arbitrarily small neighborhoods of some boundary point. The key contribution is a general theorem that enables new proofs of the absence of free subgroups in contracting groups and iterated monodromy groups of expanding maps.

ABSTRACT

We show that if a group $G$ acting faithfully on a rooted tree $T$ has a free subgroup, then either there exists a point $w$ of the boundary $\partial T$ and a free subgroup of $G$ with trivial stabilizer of $w$, or there exists $w\in\partial T$ and a free subgroup of $G$ fixing $w$ and acting faithfully on arbitrarily small neighborhoods of $w$. This can be used to prove absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki.

Motivation & Objective

• To establish a structural criterion for the existence of free subgroups in groups acting on rooted trees.
• To provide a unified framework for proving the absence of free subgroups in classes of groups acting on trees, such as contracting groups and iterated monodromy groups.
• To generalize and reprove known results on automata groups and bounded automorphisms using a topological-geometric approach.
• To formalize the intuition that 'small' action graphs on the boundary prevent free subgroups from existing.

Proposed method

• Analyzes the action of a group $ G $ on a rooted tree $ T $, focusing on stabilizers and germs at boundary points $ w \in \partial T $.
• Introduces the group of $ G $-germs $ G_{(w)} $, defined as the quotient of the stabilizer $ G_w $ by the subgroup acting trivially on a neighborhood of $ w $.
• Applies an inductive argument on the depth $ d $ of automorphisms in the class $ P_d(\mathsf{X}) $, using the fact that $ g|_v \in P_{d-1}(\mathsf{X}) $ for most $ v $.
• Uses the notion of $ g $-singular points (points with non-trivial local action) and proves they are at most countable, implying measure-zero sets.
• Applies results from Schreier graphs and amenability: if singular sets are measure-zero, Schreier graphs are amenable, which obstructs free subgroups.
• Employs a contradiction argument via a monomorphism $ \phi: \tilde{F} \to (P_{d-1}(\mathsf{X}))^U $, showing that if $ \tilde{F} $ were free, the kernel would be non-trivial, contradicting freeness.

Experimental results

Research questions

• RQ1 Under what conditions can a group acting on a rooted tree contain a free non-abelian subgroup?
• RQ2 Can the absence of free subgroups in contracting self-similar groups be formally proven using boundary dynamics?
• RQ3 How do the local actions near boundary points constrain the existence of free subgroups?
• RQ4 Can the structure of iterated monodromy groups be used to infer the absence of free subgroups?
• RQ5 What is the role of the group of germs at a boundary point in obstructing free subgroups?

Key findings

• A group $ G $ acting faithfully on a locally finite rooted tree $ T $ either has no free non-abelian subgroup, or there exists a boundary point $ w \in \partial T $ such that a free subgroup acts freely at $ w $, or a free subgroup acts faithfully on a neighborhood of $ w $.
• Contracting self-similar groups have no free subgroups, confirming a folklore conjecture.
• Iterated monodromy groups of post-critically finite rational functions and other expanding dynamical systems have no free subgroups.
• The group $ P_d(\mathsf{X}) $ of automorphisms of polynomial growth has no free subgroups, generalizing a result by S. Sidki.
• Bounded automorphisms of rooted trees generate groups without free subgroups, proven here for the first time in full generality.
• The Schreier graphs of the action on $ \partial T $ are amenable when the set of $ g $-singular points is at most countable, which helps obstruct free subgroups.