Approximations of delocalized eta invariants by their finite analogues

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 $\mathrm{\Gamma }$-regular covering space of $M$. Let $\langle \alpha \rangle$betheconjugacyclassofanon-identityelement$\alpha\in \Gamma$\mathrm{.}Suppose${\Gamma_i}$ is a sequence of finite-index normal subgroups of $\mathrm{\Gamma }$ that distinguishes $⟨\alpha ⟩$. Let ${\pi }_{{\mathrm{\Gamma }}_i}$ be the quotient map from $\mathrm{\Gamma }$ to $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_i$ and $\langle \pi_{\Gamma_i}(\alpha) \rangle$theconjugacyclassof$\pi_{\Gamma_i}(\alpha)$in$\Gamma/\Gamma_i$\mathrm{.}Ifthescalarcurvatureon$M$ 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 $M_{{\mathrm{\Gamma }}_i}$ at the conjugacy class $⟨{\pi }_{{\mathrm{\Gamma }}_i}(\alpha )⟩$, where $M_{\Gamma_i}= \widetilde M/\Gamma_i$isthefinite-sheetedcoveringspaceof$M$determinedby$\Gamma_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{{\mathrm{\Gamma }}_i}$ at the conjugacy class $⟨{\pi }_{{\mathrm{\Gamma }}_i}(\alpha )⟩$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $\mathrm{\Gamma }$.