Regularized and $L^2$-Determinants

TL;DR

This paper proves that in a tower of coverings, the regularized determinant of a generalized Laplacian converges to the $L^2$-determinant, simplifying the analysis of analytic torsion and determinants.

Contribution

It establishes the convergence of regularized determinants to $L^2$-determinants and introduces an Euler product expansion in terms of equivariant $L^2$-determinants.

Findings

Regularized determinants converge to $L^2$-determinants in tower coverings.

Provides an Euler product expansion for regularized determinants.

Shows nontriviality of analytic torsion via $L^2$-counterparts.

Abstract

It is shown that in a tower of coverings the regularized determinant of a generalized Laplacian converges to the $L^2$-determinant. This shows generic nontriviality of analytic torsion or regularized determinants since the $L^2$-counterparts are easier to compute. We further have an "Euler product expansion" for regularized determinants in terms of equivariant $L^2$-determinants.