Borel class and Cartan involution

TL;DR

This paper proves that the Borel class of certain 3-manifold group representations into PGL(n,C) remains invariant under Cartan involution up to sign, providing a shorter proof for the PGL(3,C) case.

Contribution

It offers a new, shorter proof that the Borel class is preserved under Cartan involution for these representations, extending understanding of their invariance properties.

Findings

Borel class is preserved under Cartan involution up to sign

Shorter proof for PGL(3,C) case

Extension of invariance results to PGL(n,C) representations

Abstract

In this note we prove that the Borel class of representations of 3-manifold groups to PGL(n,C) is preserved under Cartan involution up to sign. For representations to PGL(3,C) this is implied by a more general result of E. Falbel and Q. Wang, however our proof appears to be much shorter for that special case.