The Hrushovski Property for Compact Special Cube Complexes

TL;DR

This paper proves that compact nonpositively curved cube complexes can be embedded into larger complexes with extended symmetries, and under certain conditions, these embeddings preserve special properties and local isometries.

Contribution

It establishes the Hrushovski property for compact special cube complexes, extending the class of complexes with automorphism extension properties.

Findings

Any compact nonpositively curved cube complex embeds into a larger one with extended automorphisms.

Special properties can be preserved in the embedding under certain conditions.

The embedding can be made a local isometry when the complex is special and conditions are met.

Abstract

We show that any compact nonpositively curved cube complex $Y$ embeds in a compact nonpositively curved cube complex $R$ where each combinatorial injective partial local isometry of $Y$ extends to an automorphism of $R$. When $Y$ is special and the collection of injective partial local isometries satisfies certain conditions, we show that $R$ can be chosen to be special and the embedding $Y\hookrightarrow R$ can be chosen to be a local isometry.