Delocalized $L^2$-Invariants

TL;DR

This paper introduces delocalized $L^2$-invariants, extending existing invariants of closed manifolds, which can recover geometric data like the marked length spectrum of hyperbolic manifolds, highlighting their topological significance.

Contribution

It defines delocalized $L^2$-invariants based on conjugacy classes, demonstrating their topological nature and their ability to recover geometric information.

Findings

Delocalized $L^2$-invariants are constructed from conjugacy classes.

These invariants can recover the marked length spectrum of hyperbolic manifolds.

They are shown to be topological in many cases.

Abstract

We define extensions of the $L^2$-analytic invariants of closed manifolds, called delocalized $L^2$-invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spectrum of an odd-dimensional hyperbolic manifold can be recovered from its delocalized $L^2$-analytic torsion. There are technical convergence questions.