Ada Masters (University of Wollongong)
Since Connes’s 1989 paper, the use of length functions to build spectral triples for group C*-algebras has become commonplace in noncommutative geometry. This construction, although sound at the level of quantum metric spaces, always gives rise to trivial K-homology. Using ingredients from geometric group theory, this can be remedied for many CAT(0) groups, including non-discrete groups. In the process, a novel group invariant from (quantum-group-equivariant) KK-theory is uncovered. The understanding of group extensions in this framework is a microcosm of the more general problem of the constructive unbounded Kasparov product. I will also touch on the problem of building spectral triples for crossed product C*-algebras arising from dynamical systems, using the Kasparov product and new tools inspired by conformal geometry.
