Spectral triples for discrete groups
Description
The Pontrjagin dual of an Abelian discrete group is a compact topological group, but if the Abelian discrete group is finitely generated, then its dual is actually a Lie group. The reduced group $C*$-algebra of a non-Abelian discrete group is certainly a compact quantum group, but under what conditions can we endow it with the structure of a noncommutative manifold? In this talk, I'll recall Connes's construction of spectral triples for discrete groups endowed with a proper length function, and then I'll discuss how to refine this construction for discrete groups endowed with a proper array, e.g., groups with the Haagerup property together with a witnessing proper $1$-cocycle, at the price of working with unbounded $KK$-cycles. This is joint work in progress with Steve Avsec.
Spectral triples for discrete groups
Cupples I Room 115
The Pontrjagin dual of an Abelian discrete group is a compact topological group, but if the Abelian discrete group is finitely generated, then its dual is actually a Lie group. The reduced group $C*$-algebra of a non-Abelian discrete group is certainly a compact quantum group, but under what conditions can we endow it with the structure of a noncommutative manifold? In this talk, I'll recall Connes's construction of spectral triples for discrete groups endowed with a proper length function, and then I'll discuss how to refine this construction for discrete groups endowed with a proper array, e.g., groups with the Haagerup property together with a witnessing proper $1$-cocycle, at the price of working with unbounded $KK$-cycles. This is joint work in progress with Steve Avsec.