Quasi-representations of surface groups

Abstract

By a quasi-representation of a group $G$ we mean an approximately multiplicative map of $G$ to the unitary group of a unital $C^{\ast }$-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 $\Vert uv-vu\Vert$ 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 $C^{\ast }$-algebra.

25 pages; 4 figures; to appear in J. London Math. Soc

Cited by in corpus (1)

• Quasi-representations of groups and two-homology