[Paper Review] The complexity of isomorphism between countably based profinite groups

André Nies|arXiv (Cornell University)|Apr 3, 2016

Rings, Modules, and Algebras9 references3 citations

TL;DR

This paper investigates the complexity of isomorphism for countably based profinite groups using Borel reducibility in descriptive set theory. It shows that isomorphism for topologically finitely generated profinite groups is as complex as identity on the reals, while general profinite group isomorphism is complete for $S_\infty$-orbit equivalence relations, matching the complexity of isomorphism for countable graphs.

ABSTRACT

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based profinite group as a point in a Polish space. Then we study the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs.

Motivation & Objective

• To determine the descriptive set-theoretic complexity of isomorphism for countably based profinite groups.
• To classify the complexity of isomorphism using Borel reducibility, comparing it to known equivalence relations.
• To show that topologically finitely generated profinite groups have low isomorphism complexity—equivalent to identity on the reals.
• To demonstrate that general profinite group isomorphism is complete for $S_\infty$-orbit equivalence relations.
• To establish a Polish space representation of countably based profinite groups for measurable classification.

Proposed method

• Representing countably based profinite groups as points in a Polish space via inverse limits of finite quotients.
• Using Borel reducibility to compare the complexity of isomorphism to standard equivalence relations like identity on $\mathbb{R}$ and $S_\infty$-orbit equivalence.
• Applying Lubotzky’s results to show that isomorphism of topologically finitely generated profinite groups is a closed equivalence relation, hence Borel below identity on $\mathbb{R}$.
• Leveraging the universal property of the free profinite group $\widehat{F}_\omega$ to represent arbitrary profinite groups as quotients of $\widehat{F}_\omega$.
• Using the structure of the group $G = \widehat{F}_k / R$ with $R$ a closed normal subgroup to analyze centralizer sizes and commutator relations in the proof of $S_\infty$-completeness.
• Analyzing cases based on the adjacency structure of generators in the associated graph $A$ to classify centralizers and derive the isomorphism type of elements.

Experimental results

Research questions

• RQ1 What is the Borel complexity of isomorphism for topologically finitely generated profinite groups?
• RQ2 How does the complexity of isomorphism for general profinite groups compare to known equivalence relations in descriptive set theory?
• RQ3 Can countably based profinite groups be uniformly represented in a Polish space to enable measurable classification?
• RQ4 Is isomorphism of profinite groups complete for $S_\infty$-orbit equivalence relations?
• RQ5 To what extent does the topological structure of profinite groups determine their isomorphism type?

Key findings

• Isomorphism of topologically finitely generated profinite groups is Borel reducible to identity on the reals, indicating low complexity.
• The isomorphism relation for general countably based profinite groups is complete for $S_\infty$-orbit equivalence relations.
• This implies that the isomorphism problem for profinite groups is as complex as isomorphism for countable graphs, a known $S_\infty$-complete problem.
• The proof relies on analyzing centralizers of elements in a quotient of the free profinite group $\widehat{F}_\omega$ via graph-theoretic properties of generator adjacency.
• The size of centralizers in the quotient group depends on the structure of the associated graph $A$, with sizes $p-1$, $(p-1)^2$, or $p(p-1)$ depending on the case.
• In the absence of triangles and squares in the generator graph, centralizers are generated by a single element, leading to $p-1$ elements in the conjugacy class.