Quasi-representations of groups and two-homology

TL;DR

This paper generalizes the Exel-Loring formula to countable discrete groups, linking almost commuting matrices to quasi-representations and exploring invariants for quasidiagonal groups with certain geometric properties.

Contribution

It extends the Exel-Loring formula from matrices to a broad class of groups and investigates the nontriviality of related invariants in specific geometric contexts.

Findings

Generalized the Exel-Loring formula for countable groups

Established nontrivial invariants for quasidiagonal groups

Connected invariants to geometric properties like coarse embeddability

Abstract

The Exel-Loring formula asserts that two topological invariants associated to a pair of almost commuting unitary matrices coincide. Such a pair can be viewed as a quasi-representation of $\mathbb{Z}^2$. We give a generalization of this formula for countable discrete groups. We also show the nontriviality of the corresponding invariants for quasidiagonal groups which are coarsely embeddable in a Hilbert space and have nonvanishing second Betti number.