Quasi-representations of surface groups

TL;DR

This paper extends the Exel-Loring formula, relating K-theoretic invariants and winding numbers, to quasi-representations of surface groups in the context of tracial unital C*-algebras, broadening its applicability.

Contribution

It generalizes the Exel-Loring formula for quasi-representations of surface groups into the unitary group of a tracial unital C*-algebra.

Findings

Extended the Exel-Loring formula to new algebraic settings.

Connected K-theoretic invariants with winding numbers for surface groups.

Provided a broader framework for analyzing almost-commuting unitary matrices.

Abstract

By a quasi-representation of a group $G$ we mean an approximately multiplicative map of $G$ to the unitary group of a unital $C^*$-algebra. A quasi-representation induces a partially defined map at the level $K$-theory. In the early 90s Exel and Loring associated two invariants to almost-commuting pairs of unitary matrices $u$ and $v$: one a $K$-theoretic invariant, which may be regarded as the image of the Bott element in $K_0(C(\mathbb{T}^2))$ under a map induced by quasi-representation of $\mathbb{Z}^2$ in U(n); the other is the winding number in $\mathbb{C}\setminus \{0\}$ of the closed path $t\mapsto \det(tvu + (1-t)uv)$. The so-called Exel-Loring formula states that these two invariants coincide if $\|uv - vu\|$ is sufficiently small. A generalization of the Exel-Loring formula for quasi-representations of a surface group taking values in U(n) was given by the second-named author. Here we further extend this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital $C^*$-algebra.