Answering the question below, diagram groups are directed homotopy groups of directed 2-complexes (2-categories enriched over groupoids). See: Guba, V. S.; Sapir, M. V. Diagram groups and directed 2-complexes: homotopy and homology. J. Pure Appl. Algebra 205 (2006), no. 1, 1–47.
Unlike ordinary homotopy groups, these are typically non-abelian. Examples are the free groups, free Abelian groups, R. Thompson group $F$ (which is the diagram group of the dunce hat viewed as a directed 2-complex), and its relatives, as well as iterative wreath products of integers, and many other groups. Braided picture groups are defined similarly, examples are the simple R. Thompson group $V$ (again corresponds to the dunce hat). These groups do act on nice CAT(0) cubical complexes (see Farley's paper above) but this is not how they are defined. About the groups introduced by Brin and Dehornoy see the paper above.