[Paper Review] Highly faithful actions and dense free subgroups in full groups

François Le Maı̂tre|arXiv (Cornell University)|Sep 11, 2015

Advanced Operator Algebra Research13 references3 citations

TL;DR

This paper constructs highly faithful, amenable-onto-almost-every-orbit actions of the free group 𝔽ₘ on m generators that induce measure-preserving ergodic equivalence relations of cost less than m. By explicitly building topological generators for full groups and twisting them to ensure high non-freeness, the authors prove that such full groups contain dense free subgroups on m generators, strengthening Bowen's Baire category result with a constructive, explicit method.

ABSTRACT

In this paper, we show that every measure-preserving ergodic equivalence relation of cost less than m comes from a "rich" faithful invariant random subgroup of the free group on m generators, strengthening a result of Bowen which had been obtained by a Baire category argument. Our proof is completely explicit: we use our previous construction of topological generators for full groups and observe that these generators induce a totally non free action. We then twist this construction so that the action is moreover amenable onto almost every orbit and highly faithful.

Motivation & Objective

• To strengthen Bowen's Baire category result on faithful invariant random subgroups by providing an explicit, constructive proof for actions of 𝔽ₘ on equivalence relations of cost less than m.
• To classify non-free actions of 𝔽ₘ by introducing the notion of 'high faithfulness'—a strengthening of non-freeness beyond amenability or transitivity.
• To demonstrate that full groups of ergodic, measure-preserving equivalence relations of cost less than m contain dense free subgroups on m generators.
• To establish a connection between topological generation in full groups and the existence of highly faithful actions via explicit combinatorial and group-theoretic twisting.

Proposed method

• Constructs modified topological generators for the full group of a hyperfinite equivalence relation using partial isomorphisms and cycle decompositions.
• Applies a twisting procedure using a fixed partial isomorphism ψ to modify standard generators so that the resulting action becomes highly faithful.
• Uses the fact that 𝔽ₘ is residually q-finite for odd primes q to build asymptotically free actions on finite sets, which are then embedded into the full group.
• Employs the closure properties of full groups and the structure of joinings of equivalence relations to show that the generated group is dense in the full group.
• Leverages the fact that disjoint supports and specific cycle structures allow the closure of certain products to generate key elements like U and C_{Φ̃ᵢ}.
• Applies Theorem 4.5 to induce a new 𝔽ₘ-action on a subset A with finite orbits, ensuring high faithfulness and amenability onto almost every orbit.

Experimental results

Research questions

• RQ1 Can one construct an explicit, highly faithful action of 𝔽ₘ on a measure-preserving equivalence relation of cost less than m, beyond the existence guaranteed by Baire category arguments?
• RQ2 To what extent can non-freeness of 𝔽ₘ-actions be strengthened to classify low-cost equivalence relations?
• RQ3 Does the full group of a cost-less-than-m ergodic equivalence relation contain a dense free subgroup on m generators, and if so, how can such a subgroup be explicitly realized?
• RQ4 Can amenability onto almost every orbit be combined with high faithfulness in a single action of 𝔽ₘ?
• RQ5 How can topological generators of full groups be modified to ensure both high faithfulness and density in the full group?

Key findings

• Every measure-preserving ergodic equivalence relation of cost less than m arises from a highly faithful action of the free group on m generators.
• The full group of such an equivalence relation contains a dense free subgroup on m generators, which is explicitly constructed.
• The constructed action is amenable onto almost every orbit, meaning each orbit's group action is amenable, a strong form of non-freeness.
• The action is highly faithful, meaning the action of the free product ℤ * 𝔽ₘ is faithful on a conull set, ensuring strong non-freeness.
• The proof is fully explicit and constructive, avoiding Baire category arguments used in earlier results.
• The method uses finite-order permutations and cycle structures to ensure that the closure of certain products generates the full group.