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 by examining one hyperplane at a time. Specifically, we prove that there does not exist an algorithm that, given a compact non-positively curved squared 2-complex $X$ and a hyperplane $H$ in $X$, will decide whether or not there is a finite-sheeted cover of $X$ in which no lift of $H$ self-osculates.
