Representations of 3-manifold groups in PGL(n,C) and their restriction to the boundary

TL;DR

This paper extends the study of 3-manifold group representations from PGL(2,C) to PGL(n,C), developing a combinatorial approach that leads to new rigidity results and generalizes existing theories.

Contribution

It generalizes the gluing equations and symplectic structures from PGL(2,C) to PGL(n,C), providing a new combinatorial proof of local rigidity for these representations.

Findings

Established a PGL(n,C) analog of the gluing equations

Revealed a symplectic structure similar to the PGL(2,C) case

Proved local rigidity results for PGL(n,C) representations

Abstract

We study here the space of representations of a fundamental group of a 3-manifold into PGL(n,C). Thurston, Neumann and Zagier initiated a strategy (in the case of PGL(2,C)) consisting in: triangulate the manifold, assign shapes to each pieces and then try to glue back. This leads to the "gluing equations" and the Neumann-Zagier symplectic space. Building on the works of Dimofte-Gabella-Goncharov and Bergeron-Falbel-Guilloux, we complete the picture in the case of PGL(n,C). We recover a situation very similar to the case of PGL(2,C). This allows for example to obtain a combinatorial proof of a local rigidity results for such representations.