21.124 (2026)

A group $G$ is said to be virtually special if $G$ has a finite-index subgroup isomorphic to the fundamental group of a special complex (in the sense of F. Haglund, D. T. Wise, Geom. Funct. Anal., 17, no. 5 (2008), 1551–1620; cf 20.60.) A group $G$ is called a $\text{CAT}(0)$ group if it acts properly discontinuously and cocompactly by isometries on a $\text{CAT}(0)$ metric space.
$\qquad$ a) Is every $\text{CAT}(0)$ free-by-cyclic group virtually special?
$\qquad$ b) A weaker question: does every $\text{CAT}(0)$ free-by-cyclic group virtually embed into a right-angled Artin group?