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Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism

Gianluca Paolini, Andre Nies

TL;DR

The paper investigates the automorphism and outer automorphism groups of oligomorphic, Roelcke precompact subgroups of Sym(ω), proving Inn(G) is closed in Aut(G) and Out(G) is t.d.l.c. It develops a model-theoretic framework using canonical structures M_G, imaginaries, and orbital structures to link open subgroups to definable data, yielding a profinite description of N_G/G and a Borel pathway to smooth isomorphism classifications. A central general theorem provides a criterion for smoothness of the isomorphism relation on Borel classes of oligomorphic groups via a P-subgroup-based structure C_G^P, with two concrete smooth classes (no algebraicity and finitely many essential subgroups). The paper further shows that for these classes, Aut(G) is itself oligomorphic (up to isomorphism) and Out(G) is profinite, and discusses weak elimination of imaginaries as a pathway to smoothness, connecting model-theoretic properties to topological group structure. Overall, it advances understanding of the complexity of classifying oligomorphic groups and links algebraic, topological, and model-theoretic dimensions through a unified framework.

Abstract

The paper follows two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\Sym(ω)$ of permutations of the natural numbers. Then $\Inn(G)$ is closed in $\Aut(G)$, where $\Aut(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $N_G/G$ is profinite, where $N_G$ denotes the normaliser of~$G$ in $\Sym(ω)$, and the topological group $\Out(G)= \Aut(G)/\Inn(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\Aut(G)$ is topologically isomorphic to an oligomorphic group, and $\Out(G)$ is profinite.

Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism

TL;DR

The paper investigates the automorphism and outer automorphism groups of oligomorphic, Roelcke precompact subgroups of Sym(ω), proving Inn(G) is closed in Aut(G) and Out(G) is t.d.l.c. It develops a model-theoretic framework using canonical structures M_G, imaginaries, and orbital structures to link open subgroups to definable data, yielding a profinite description of N_G/G and a Borel pathway to smooth isomorphism classifications. A central general theorem provides a criterion for smoothness of the isomorphism relation on Borel classes of oligomorphic groups via a P-subgroup-based structure C_G^P, with two concrete smooth classes (no algebraicity and finitely many essential subgroups). The paper further shows that for these classes, Aut(G) is itself oligomorphic (up to isomorphism) and Out(G) is profinite, and discusses weak elimination of imaginaries as a pathway to smoothness, connecting model-theoretic properties to topological group structure. Overall, it advances understanding of the complexity of classifying oligomorphic groups and links algebraic, topological, and model-theoretic dimensions through a unified framework.

Abstract

The paper follows two interconnected directions. 1. Let be a Roelcke precompact closed subgroup of the group of permutations of the natural numbers. Then is closed in , where carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~ is oligomorphic, is profinite, where denotes the normaliser of~ in , and the topological group is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if is in such a Borel class, then is topologically isomorphic to an oligomorphic group, and is profinite.
Paper Structure (12 sections, 17 theorems, 15 equations)

This paper contains 12 sections, 17 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. $\mathrm{Aut}(G)$ for Roelcke precompact $G$
  3. Model-theoretic concepts, with applications
  4. Imaginaries and $M^{\mathrm{eq}}$
  5. Orbital structures and profinite groups
  6. Open cosets and pairs of imaginaries
  7. $\mathrm{Out}(G)$ is t.d.l.c. for oligomorphic groups $G$
  8. A general result involving a Borel property of open subgroups
  9. Two smooth classes
  10. Groups with no algebraicity
  11. Finitely many essential subgroups up to conjugacy
  12. Smoothness for weak elimination of imaginaries

Key Result

Theorem 1.1

Let $G \leqslant \mathrm{Sym}(\omega)$ be an oligomorphic group. Let $N_G$ denote its normaliser in $\mathrm{Sym}(\omega)$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.3
  • Definition 2.5
  • proof
  • Theorem 2.7
  • proof
  • Remark 2.8
  • Example 2.9: Similar to Example 4.5 in Wu:71
  • ...and 47 more