Eta-Invariants for Parameter-Dependent Operators Associated with an Action of a Discrete Group

\(\eta\)-invariants for a class of parameter-dependent nonlocal operators associated with an isometric action of a discrete group of polynomial growth on a smooth closed manifold are studied. The \(\eta\)-invariant is defined as the regularization of the winding number. The formula for the variation of the \(\eta\)-invariant when the operator changes is obtained. The results are based on the study of asymptotic expansions of traces of parameter-dependent nonlocal operators.