A finitedimensional version of Fredholm representations

TL;DR

This paper explores how pairs of near-homomorphisms from a discrete group to the unitary group, called balanced pairs, relate to K-theory, and how Fredholm representations can generate such pairs.

Contribution

It introduces the concept of balanced pairs of maps and establishes their connection to K-theory, extending the understanding of Fredholm representations in this context.

Findings

Balanced pairs determine elements in K-theory groups.

Fredholm representations can generate balanced pairs.

The approach links near-homomorphisms to topological invariants.

Abstract

We consider pairs of maps from a discrete group to the unitary group. The deficiencies of these maps from being homomorphisms may be great, but if they are close to each other then we call such pairs balanced. We show that balanced pairs determine elements in the K-theory group of the classifying space of the discrete group. We also show that a Fredholm representation determines balanced pairs.