Higher Lefschetz formulas on Γ-proper manifolds

Abstract

Let $\Gamma$ be a finitely generated discrete group acting properly and cocompactly on a smooth manifold M. By employing heat-kernel techniques we prove a geometric formula for the pairing of the index class associated to a $\Gamma$-equivariant Dirac operator $D$ with a delocalized cyclic cocycles $\tau$ in $H{P}^{\bullet }(\mathbb{C}\Gamma ,⟨\gamma ⟩)$. Our formula takes place on the fixed point manifold $M^{\gamma }$ and should be regarded as a higher Lefschetz formula for $D$. The formula involves the Atiyah-Segal-Singer form and an explicit $Z_{\gamma }$-invariant form on $M^{\gamma }$ that is naturally associated to $\tau \in H{P}^{\bullet }(\mathbb{C}\Gamma ,⟨\gamma ⟩)$

64 pages. An improvement of the main theorem is presented. In particular, the exponential growth assumption on the group cocycles is dropped