[Paper Review] Explicit examples of equivalence relations and II$_1$ factors with prescribed uncountable fundamental group

Steven Deprez|arXiv (Cornell University)|Oct 18, 2010

Advanced Topology and Set Theory3 citations

TL;DR

This paper constructs explicit examples of equivalence relations and II$_1$ factors with prescribed fundamental groups, specifically any countable or certain uncountable subgroups of $\mathbb{R}_+$. It provides a concrete realization of previously existential results by Popa and Vaes, using a direct, non-generic method to achieve the desired fundamental group structure.

ABSTRACT

Given a subgroup $\mathcal{F}$ of $\mathbb{R}_+$, that is either countable or belongs to a large class of uncountable subgroups, we construct explicit examples of equivalence relations $\mathcal{R}$ and II$_1$ factors $M$ that have fundamental group $\mathcal{F}$. The existence of such examples was proven earlier by Popa and Vaes using a Baire category argument.

Motivation & Objective

• To provide explicit constructions of equivalence relations and II$_1$ factors with fundamental groups equal to a given subgroup $\mathcal{F} \subset \mathbb{R}_+$, rather than relying on non-constructive Baire category arguments.
• To extend the known existence results of II$_1$ factors with arbitrary fundamental groups to include uncountable subgroups of $\mathbb{R}_+$, particularly those in a large class of such groups.
• To resolve the gap in the literature by offering concrete, computable examples where the fundamental group is not just proven to exist but is explicitly realized.

Proposed method

• The construction employs a direct, combinatorial approach to build equivalence relations $\mathcal{R}$ with the desired fundamental group $\mathcal{F}$, avoiding probabilistic or generic existence proofs.
• It uses the theory of group measure space constructions to associate II$_1$ factors $M$ to the equivalence relations $\mathcal{R}$, ensuring the fundamental group of $M$ matches $\mathcal{F}$.
• The method relies on analyzing the structure of the orbit equivalence relation induced by a suitable group action on a probability space, tailored to generate the target fundamental group.
• Key techniques involve manipulating the scaling properties of the equivalence relation through controlled group actions and measure-theoretic constraints.
• The construction is designed to work uniformly across a broad class of uncountable subgroups of $\mathbb{R}_+$, not just countable ones.
• It ensures that the fundamental group of the resulting II$_1$ factor is exactly the prescribed subgroup $\mathcal{F}$, without over-approximation or genericity.

Experimental results

Research questions

• RQ1 Can explicit examples of II$_1$ factors with a prescribed uncountable fundamental group be constructed, rather than just proven to exist?
• RQ2 What class of uncountable subgroups of $\mathbb{R}_+$ can be realized as the fundamental group of a II$_1$ factor via a constructive method?
• RQ3 How can the fundamental group of an equivalence relation be controlled and realized explicitly through group actions and measure space constructions?
• RQ4 Is it possible to bypass the Baire category argument used by Popa and Vaes to achieve explicit, non-generic constructions of such factors?

Key findings

• The paper constructs explicit equivalence relations $\mathcal{R}$ whose fundamental group is any given countable subgroup of $\mathbb{R}_+$.
• It provides explicit II$_1$ factors $M$ whose fundamental group is any given uncountable subgroup of $\mathbb{R}_+$, provided the subgroup belongs to a large, explicitly defined class.
• The constructions are not based on generic or measure-theoretic existence arguments but are fully explicit and combinatorially realizable.
• The fundamental group of the resulting II$_1$ factor is exactly the prescribed subgroup $\mathcal{F}$, with no ambiguity or over-approximation.
• The method applies uniformly to both countable and uncountable subgroups, unifying the construction across different cases.