[Paper Review] Coherent extension of partial automorphisms, free amalgamation, and automorphism groups

Daoud Siniora, Sławomir Solecki|arXiv (Cornell University)|May 4, 2017

Advanced Topology and Set Theory18 references4 citations

TL;DR

This paper introduces coherent extension of partial automorphisms as a strengthened version of the Herwig–Lascar and Hodkinson–Otto extension theorems, establishing coherent EPPA for Fraïssé classes of $φ$-free structures and free homogeneous structures over finite relational languages. The key contribution is proving that the automorphism groups of such structures—like the rational Urysohn space and free homogeneous structures—contain dense locally finite subgroups and admit ample generics, with implications for topological group properties such as the small index property and Bergman's property.

ABSTRACT

We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraisse classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraisse limit of any Fraisse class that is the class of all $\mathcal{F}$-free structures (in the Herwig--Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language, all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

Motivation & Objective

• To strengthen the Herwig–Lascar and Hodkinson–Otto extension theorems by introducing coherent EPPA, a notion that preserves composition structure in automorphism extensions.
• To establish that Fraïssé classes of $φ$-free structures and free homogeneous structures over finite relational languages satisfy coherent EPPA.
• To deduce topological group-theoretic consequences, including the existence of dense locally finite subgroups and ample generics in automorphism groups.
• To connect coherent EPPA to the small index property and other structural properties of automorphism groups of homogeneous structures.

Proposed method

• Introduce the notion of coherent maps between families of partial bijections, ensuring that image triples of coherent triples remain coherent.
• Reorganize and enrich the original proofs of the Herwig–Lascar and Hodkinson–Otto theorems with new techniques, including the use of Mackey range from ergodic theory.
• Apply coherent EPPA to Fraïssé classes of $φ$-free structures and free homogeneous structures, proving that such classes admit coherent EPPA-extensions.
• Use the existence of coherent EPPA to construct dense locally finite subgroups in automorphism groups of homogeneous structures.
• Leverage the amalgamation base condition and homogeneity to apply results from [9] and [6] to establish ample generics.
• Apply group-theoretic results from [10] to deduce the small index property, uncountable cofinality, Bergman’s property, and Serre’s property (FA) for the automorphism groups.

Experimental results

Research questions

• RQ1 Can the extension property for partial automorphisms (EPPA) be strengthened to a coherent version that preserves composition structure in automorphism extensions?
• RQ2 Does the coherent EPPA property hold for Fraïssé classes of $φ$-free structures and free homogeneous structures over finite relational languages?
• RQ3 Does coherent EPPA imply the existence of a dense locally finite subgroup in the automorphism group of a homogeneous structure?
• RQ4 Can ample generics be established for automorphism groups of free homogeneous structures using coherent EPPA?
• RQ5 What topological and group-theoretic properties (e.g., small index property, Bergman’s property) follow from coherent EPPA and ample generics?

Key findings

• The automorphism group of the rational Urysohn space contains a dense locally finite subgroup.
• The automorphism group of any Fraïssé limit of a $φ$-free class (in the Herwig–Lascar sense) contains a dense locally finite subgroup.
• The automorphism group of any free homogeneous structure over a finite relational language contains a dense locally finite subgroup.
• Any free homogeneous structure over a finite relational language admits ample generics.
• The automorphism group of any free homogeneous structure over a finite relational language has the small index property, uncountable cofinality, 21-Bergman property, and Serre’s property (FA).