Nonpositively curved cube complexes were introduced by Gromov in his 1987 essay on “hyperbolic groups” which founded the field “geometric group theory”.
While Gromov suggested them as a simple source of higher dimensional ex- amples, nonpositively curved cube complexes have emerged as one of the central objects in the field. A considerable portion of combinatorial group theory owes its soul to hidden nonpositively curved cube complexes. Indeed many ideas related to splittings of groups or combinatorial rewording can be reinterpreted elegantly us- ing cube complexes. Moreover, nonpositively curved cube complexes have become increasingly important in parts of geometric group theory, and provide a natural stage for the study of hyperbolic 3-manifolds, most arithmetic hyperbolic lattices, artin and coxeter groups, small-cancellation groups, the JSJ splitting, codimension- 1 subgroups, median spaces, spaces with walls, and a(T)menability - the “opposite” of Kazhdan’s property (T).
Nonpositively curved cube complexes (and the 1-skeletons of their CAT(0) universal covers) have appeared in other places and in other guises in mathematics, for instance: Robot configuration spaces, Phylogenetic Trees, and as Median Graphs.
We will spend the semester learning about these objects and their essential properties: combinatorial, structural, and metric, and their relations to and context within geometric group theory.