  We prove the Tits Alternative for groups acting on $2$-dimensional
"recurrent" complexes with uniformly bounded cell stabilisers. This class of
complexes includes, among others: $2$-dimensional Euclidean buildings,
$2$-dimensional systolic complexes, $B(6)$-small cancellation complexes, and
standard Cayley complexes for Artin groups of extra-large type. In the appendix
written jointly with Jon McCammond we extend the result to a class of
$2$-dimensional Artin groups containing all large-type Artin groups.
