Commensurability and the character variety

Commensurability and the character variety

Recall that hyperbolic 3-manifolds M and N are said to be commensurable if they have a common finite sheeted covering. This is equivalent to the fundamental groups having subgroups of finite index which are conjugate in PSL(2,C). In general it is very difficult to determine if two manifolds are commensurable or not, once the most obvious invariants of commensurability (for example, the invarian...

Towards the boundary of the character variety

Detecting Essential Surfaces as Intersections in the Character Variety

We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are non-integral and detect a Seifert surface.

Sl2(c)-character Variety of a Hyperbolic Link and Regulator

In this paper, we study the SL2(C) character variety of a hyperbolic link in S. We analyze a special smooth projective variety Y h arising from some 1-dimensional irreducible slices on the character variety. We prove that a natural symbol obtained from these 1-dimensional slices is a torsion in K2(C(Y )). By using the regulator map from K2 to the corresponding Deligne cohomology, we get some va...

Traces, lengths, axes and commensurability

This paper is based on three lectures given by the author at the workshop, “Hyperbolic geometry and arithmetic: a crossview” held at The Université Paul Sabatier, Toulouse in November 2012. The goal of the lectures was to describe recent work on the extent to which various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. This ...