The Tits Alternative for groups acting properly on 2-dimensional CAT(0) complexes

A group is said to satisfy the Tits Alternative if any of its finitely generated subgroups either contains a free nonabelian group or is virtually (that is, has a finite index subgroup which is) solvable. Intuitively the alternative says that every such subgroup is either (respectively) "very big" or "very small". It is believed that groups acting properly on nonpositively curved spaces have this property. One important notion of nonpositive curvature for metric spaces is the CAT(0) property. Trees are 1-dimensional CAT(0) spaces, and it is relatively easy to show that groups acting properly on trees satisfy the Tits Alternative. For higher dimensions, the problem has been open. Together with Piotr Przytycki we showed that groups acting properly on 2-dimensional CAT(0) complexes satisfy the Tits Alternative. In particular, this proves the Tits Alternative for a few classical families of groups, including some Artin groups and some automorphism groups of affine spaces coming from algebraic geometry. I will present the result, its motivations, consequences, and elements of the proof.