L2 dimensions of spaces of braid-invariant harmonic forms

TL;DR

This paper computes the von Neumann dimensions of L2 spaces of braid-invariant harmonic forms on product manifolds with group actions, extending the understanding of harmonic analysis in geometric group theory.

Contribution

It introduces a method to calculate braided L2-Betti numbers for spaces with braid group symmetries, linking harmonic forms and von Neumann dimensions.

Findings

Explicit formulas for braided L2-Betti numbers

Connection between harmonic forms and group actions

Extension of L2 harmonic analysis to braid-invariant contexts

Abstract

Let X be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L2 spaces of harmonic vector-valued forms on the product manifold X^N, which are invariant with respect to an action of the braid group B_N, and compute their von Neumann dimensions (the braided L2- Betti numbers)