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 $\widetilde M$ is a $Γ$-regular covering space of $M$. Let $\langle α\rangle$ be the conjugacy class of a non-identity element $α\in Γ$. Suppose $\{Γ_i\}$ is a sequence of finite-index normal subgroups of $Γ$ that distinguishes $\langle α\rangle$. Let $π_{Γ_i}$ be the quotient map from $Γ$ to $Γ/Γ_i$ and $\langle π_{Γ_i}(α) \rangle$ the conjugacy class of $π_{Γ_i}(α)$ in $Γ/Γ_i$. 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 $\widetilde M$ at the conjugacy class $\langle α\rangle$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{Γ_i}$ at the conjugacy class $\langle π_{Γ_i}(α) \rangle$, where $M_{Γ_i}= \widetilde M/Γ_i$ is the finite-sheeted covering space of $M$ determined by $Γ_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{Γ_i}$ at the conjugacy class $\langle π_{Γ_i}(α) \rangle$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $Γ$.