The rational Gurarii space and its linear isometry group

TL;DR

This paper investigates the structure of the rational Gurarii space, showing limitations in its isometry group and properties of finite-dimensional polyhedral spaces, with implications for their automorphism groups.

Contribution

It establishes that the classes of finite-dimensional polyhedral and rational polyhedral spaces lack the weak amalgamation and Hrushovski properties, affecting the structure of the Gurarii space's isometry group.

Findings

The classes of partial isometries lack the weak amalgamation property.

The rational Gurarii space's isometry group does not have a comeager conjugacy class.

Finite-dimensional polyhedral spaces lack the Hrushovski property.

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.