Chern Characters of Bundles Associated to Almost Representations of Discrete Groups

A group is said to be matricially stable if every function from the group to unitary matrices that is "almost multiplicative" in the point-operator norm topology is "close," in the same topology, to a genuine representation. A result of Dadarlat shows that even cohomology obstructs matricial stability. The obstruction in his proof can be realized as follows. To each almost-representation,  a vector bundle may be associated. This vector bundle has topological invariants, called Chern characters, which lie in the even cohomology of the group. If any of these invariants are nonzero, the almost-representation is far from a genuine representation. The first Chern character can be computed with the "winding number argument" of Kazhdan, Exel, and Loring, but the other invariants are harder to compute explicitly. In this talk, Professor Forrest Glebe will discuss results that allow the computation of higher invariants in specific cases: when the failure to be multiplicative is scalar (joint work with Marius Dadarlat) and when the failure to be multiplicative is small in a Schatten p-norm.