Quasi-representations of surface groups (Q2854252)

A quasi-representation of a group \(G\) is an approximately multiplicative map of \(G\) to the unitary group of a unital \(C^*\)-algebra. Quasi-representations are of particular interest since they can be used to detect \(K\)-theory. In fact, each quasi-representation induces a partially defined map at the level of \(K\)-theory. In [J. Funct. Anal. 95, No.~2, 364--376 (1991; Zbl 0748.46031)], \textit{R. Exel} and \textit{T. A. 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 a quasi-representation of \(\mathbb{Z}^2\) in \(U(n)\); the other is the winding number in \(\mathbb{C}\backslash\{0\}\) 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 in [\textit{M. Dadarlat}, J. Topol. Anal. 4, No.~3, 297--319 (2012; Zbl 1258.46029)].NEWLINENEWLINE The paper under review extends this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital \(C^*\)-algebra. To be a bit more precise, in Section~2, the authors start by defining quasi-representations and the invariants which they intend to study, and state the main result, Theorem 2.6. These invariants make use of the Mishchenko line bundle, which is therefore discussed in Section~3. The push-forward of this bundle by a quasi-representation is investigated in Section~4. Section~5 contains the main technical result, Theorem 5.4, which computes one of their invariants in terms of the de la Harpe-Skandalis determinant. To obtain the formula given in Theorem 2.6, the authors work with concrete triangulations of oriented surfaces; this is the content of Section~6. Assembling these results in Section~7 finally yields a proof of Theorem 2.6.