REPRESENTATIONS OF 3-MANIFOLDS GROUPS IN PGL(n, C) AND THEIR RESTRICTION TO THE BOUNDARY

Antonin Guilloux (1, 2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 UPMC - Université Pierre et Marie Curie - Paris 6

Let M be a cusped 3-manifold – e.g. a knot complement – and note ∂M the collection of its peripheral tori. Thurston [Thu79] gave a combinatorial way to produce hyperbolic structures via triangulation and the so-called gluing equations. This gives coordinates on the space of representations of π 1 (M) to PGL(2, C). In their paper [NZ85], Neumann and Zagier showed how this coordinates are adapted to describe this space of representations as a totally isotropic subvariety lying inside a space equipped with a 2-form – now called Neumann-Zagier symplectic space. And they related this 2-form with a natural symplectic form on the space of representations of π(∂M) to PGL(2, C): the Weil-Petersson form. Subsequent works of Neumann [Neu92] and Kabaya [Kab07] extended the scope of the previous works. We fulfill here, as Garoufalidis-Zickert [GZ13], the generalization of these works to representations to PGL(3, C).

Domaines: Topologie géométrique [math.GT]

Antonin Guilloux. REPRESENTATIONS OF 3-MANIFOLDS GROUPS IN PGL(n, C) AND THEIR RESTRICTION TO THE BOUNDARY. 2016. ⟨hal-01370280⟩