The $\ell^\infty$-semi-norm on uniformly finite homology

TL;DR

This paper investigates the $\, ext{ extlbrackdbl}\, ext{ extperthousand}\, ext{ extbrackdbl}\,$-semi-norm in uniformly finite homology, revealing its non-triviality in degree 0 with integral coefficients and triviality in higher degrees with real coefficients, impacting rigidity results.

Contribution

It introduces a new perspective on the $\, ext{ extlbrackdbl}\, ext{ extperthousand}\, ext{ extbrackdbl}\,$-semi-norm in uniformly finite homology, linking it to rigidity and establishing its behavior across degrees and coefficients.

Findings

Semi-norm allows a new formulation of Whyte's rigidity in degree 0 with integer coefficients.

Semi-norm is trivial in higher degrees with real coefficients.

Results differentiate behavior of the semi-norm across degrees and coefficient types.

Abstract

Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical $\ell^\infty$-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in degree 0 with integral coefficients allows for a new formulation of Whyte's rigidity result. In contrast, we prove that this semi-norm is trivial on uniformly finite homology in higher degrees with real coefficients.