Approximations of delocalized eta invariants by their finite analogues

Abstract

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $\tilde{M}$ is a $\Gamma$-regular covering space of $M$. Let $⟨\alpha ⟩$ be the conjugacy class of a non-identity element $\alpha \in \Gamma$. Suppose $\{{\Gamma }_i\}$ is a sequence of finite-index normal subgroups of $\Gamma$ that distinguishes $⟨\alpha ⟩$. Let ${\pi }_{{\Gamma }_i}$ be the quotient map from $\Gamma$ to $\Gamma /{\Gamma }_i$ and $⟨{\pi }_{{\Gamma }_i}(\alpha )⟩$ the conjugacy class of ${\pi }_{{\Gamma }_i}(\alpha )$ in $\Gamma /{\Gamma }_i$. If the scalar curvature on $M$ is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of $\tilde{M}$ at the conjugacy class $⟨\alpha ⟩$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{{\Gamma }_i}$ at the conjugacy class $⟨{\pi }_{{\Gamma }_i}(\alpha )⟩$, where $M_{{\Gamma }_i}=\tilde{M}/{\Gamma }_i$ is the finite-sheeted covering space of $M$ determined by ${\Gamma }_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{{\Gamma }_i}$ at the conjugacy class $⟨{\pi }_{{\Gamma }_i}(\alpha )⟩$ converges, under the assumption that the rational maximal Baum–Connes conjecture holds for $\Gamma$.