Analytic torsion on manifolds under locally compact group actions

Abstract

For a complete Riemannian manifold without boundary which a unimodular locally compact group properly cocompact acts on it, under some conditions, we define and study the analytic torsion on it by using the $G$-trace defined in ($L^{2}$-index formula for proper cocompact group actions, preprint). For a fiber bundle $\pi:M\to B$, if there is a unimodular locally compact group acts fiberwisely properly and cocompact on it, we define the torsion form for it, and show that the zero degree part of the torsion form is the analytic torsion. This can be viewed as an extension of the $L^{2}$-analytic torsion.