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 $C^\ast$-algebra $C^\ast_r\Gamma$ of a discrete group $\Gamma$ is certainly a compact quantum group, and by a result of Connes, a proper length function on $\Gamma$ suffices to define a metrically non-trivial spectral triple for $C^\ast_r\Gamma$. In this talk, I’ll discuss a refinement of this construction for a-T(T)-menable discrete groups that directly extends Pontrjagin duality for finitely generated discrete Abelian groups, but at the price of working with unbounded $KK$-cycles.

This is joint work in progress with Steve Avsec.