# Virtual Specialness of certain graphs of special cube complexes

## TL;DR

This paper studies when certain complex cube structures are virtually special, showing that under specific hyperbolic and structural conditions, these complexes have desirable properties related to their fundamental groups.

## Contribution

It proves the virtual specialness of compact cube complexes split as graphs of nonpositively curved complexes under hyperbolic and finite stature conditions.

## Key findings

- Virtual specialness holds when vertex groups are hyperbolic and the complex has finite stature.
- Generalizes results from tree times tree lattice cases.
- Provides conditions for virtual specialness in complex cube structures.

## Abstract

We investigate the virtual specialness of a compact cube complex $X$ that splits as a graph of nonpositively curved cube complexes. We prove virtual specialness of $X$ when each vertex space of $X$ has word-hyperbolic $\pi_1$ and $\pi_1X$ has ``finite stature'' relative to its edge groups. The results generalize the motivating case when tree $\times$ tree lattices are virtual products.