On the recognition problem for virtually special cube complexes

TL;DR

This paper proves that determining whether a finite non-positively curved cube complex is virtually special cannot be decided by examining individual hyperplanes, highlighting the problem's inherent computational complexity.

Contribution

It establishes the non-existence of an algorithm to decide virtual specialness locally via hyperplane analysis in cube complexes.

Findings

No local decision procedure exists for virtual specialness.

The problem is undecidable when analyzing hyperplanes individually.

Highlights limitations of algorithmic approaches in geometric group theory.

Abstract

We address the question of whether the property of being virtually special (in the sense of Haglund and Wise) is algorithmically decidable for finite, non-positively curved cube complexes. Our main theorem shows that it cannot be decided locally, i.e. by examining one hyperplane at a time. Specifically, we prove that there does not exist an algorithm that, given a compact non-positively squared 2-complex X and a hyperplane H in X can decide whether or not there is a finite-sheeted cover of X in which no lift of H self-osculates.