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

<span class="mathjax-tex">\(\eta\)</span>-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 <span class="mathjax-tex">\(\eta\)</span>-invariant is defined as the regularization of the winding number. The formula for the variation of the <span class="mathjax-tex">\(\eta\)</span>-invariant when the operator changes is obtained. The results are based on the study of asymptotic expansions of traces of parameter-dependent nonlocal operators.