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The rational Gurarii space and its linear isometry group

Ondřej Kurka, Maciej Malicki

TL;DR

The paper analyzes amalgamation properties of partial isometries in finite-dimensional polyhedral spaces, showing that both rational and real polyhedral classes fail the Hrushovski property and that their partial-automorphism classes fail the weak amalgamation property. These obstructions imply that the linear isometry group of the rational Gurarii space, Aut( G_Q ), has no comeager conjugacy class, contrasting with the ample generics observed in related linear classes. The authors employ Fraïssé theory, polyhedral geometry, and a technical Key Lemma (constructed via shift spaces and quotients) to exhibit obstructions and to relate these properties to automatic continuity and ample generics. Overall, the work delineates fundamental limits of extending Sabok's approach to the Gurarii setting and highlights distinct behavior between rational and full Gurarii spaces.

Abstract

We show that the classes of partial isometries in finite-dimensional polyhedral spaces and in finite-dimensional rational polyhedral spaces do not have the weak amalgamation property. This implies that the linear isometry group of the rational Gurarii space does not have a comeager conjugacy class. Our methods demonstrate also that the classes of finite-dimensional polyhedral spaces and of finite-dimensional rational polyhedral spaces fail to have the Hrushovski property.

The rational Gurarii space and its linear isometry group

TL;DR

The paper analyzes amalgamation properties of partial isometries in finite-dimensional polyhedral spaces, showing that both rational and real polyhedral classes fail the Hrushovski property and that their partial-automorphism classes fail the weak amalgamation property. These obstructions imply that the linear isometry group of the rational Gurarii space, Aut( G_Q ), has no comeager conjugacy class, contrasting with the ample generics observed in related linear classes. The authors employ Fraïssé theory, polyhedral geometry, and a technical Key Lemma (constructed via shift spaces and quotients) to exhibit obstructions and to relate these properties to automatic continuity and ample generics. Overall, the work delineates fundamental limits of extending Sabok's approach to the Gurarii setting and highlights distinct behavior between rational and full Gurarii spaces.

Abstract

We show that the classes of partial isometries in finite-dimensional polyhedral spaces and in finite-dimensional rational polyhedral spaces do not have the weak amalgamation property. This implies that the linear isometry group of the rational Gurarii space does not have a comeager conjugacy class. Our methods demonstrate also that the classes of finite-dimensional polyhedral spaces and of finite-dimensional rational polyhedral spaces fail to have the Hrushovski property.

Paper Structure

This paper contains 5 sections, 8 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Fraïssé classes
  3. Polyhedral spaces
  4. Two classes of linear spaces
  5. Key lemma

Key Result

Theorem 2.1

Let $M$ be the Fraïssé limit of a Fraïssé class $\mathcal{K}$, and let $n \geq 1$. The group $\hbox{Aut}(M)$ has a comeager diagonal $n$-conjugacy class iff $\mathcal{K}_n$ has JEP and WAP.

Theorems & Definitions (27)

  • Theorem 2.1
  • proof
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 17 more