Resumen de Eigenvalue varieties of abelian trees of groups and link-manifolds
François Malabre
The A-polynomial of a knot in S^3 is a two variable polynomial obtained by projecting the SL(2,C)-character variety of the knot-group to the character variety of its peripheral sub- group. It distinguishes the unknot and detects some boundary slopes of essential surfaces in knot exteriors.
The notion of A-polynomial has been generalized to 3-manifolds with non-connected toric boundaries; if M is a 3-manifold bounded by n tori, this produces an algebraic subset E(M) of C^{2n} called the eigenvalue variety of M. It has dimension at most n and still detects systems of boundary slopes of surfaces in M .
The eigenvalue variety of M always contains a part Er(M) arising from reducible characters and with maximal dimension. If M is hyperbolic, E(M) contains another top- dimensional component; for which 3-manifolds is this true remains an open question.
In this thesis, this matter is studied for two families of 3-manifolds with toric bound- aries and, via two very different technics, we provide a positive answer for both cases.
On the one hand, we study Brunnian links in S3, links in the standard 3-sphere for which any strict sublink is trivial. Using special properties of these links and stability under certain Dehn fillings we prove that, if M is the exterior of a Brunnian link different from the trivial link or the Hopf link, then E(M) admits a top-dimensional component different from Er(M). This is achieved generalizing the technic applied to knots in S^3, using Kronheimer-Mrowka theorem.
On the other hand, we consider a family of link-manifolds, exteriors of links in integer- homology spheres. Link-manifolds are equipped with standard peripheral systems of meridians and longitudes and are stable under splicing, gluing two link-manifolds along respective boundary components, identifying the meridian of each side to the longitude of the other. This yields a well-defined notion of torus decomposition and a link-manifold is called a graph link-manifold if there exists such a decomposition for which each piece is Seifert-fibred. Discarding trivial cases, we prove that all graph link-manifolds produce another top-dimensional component in their eigenvalue variety.
For this second proof, we propose a further generalization of the eigenvalue variety that also takes into account internal tori and this is introduced in the broader context of abelian trees of groups. A tree of group is called abelian if all its edge groups are commutative; in that case, we define the eigenvalue variety of an abelian tree of groups, an algebraic variety compatible with two natural operations on trees: merging and contraction. This enables to study the eigenvalue variety of a link-manifold through the eigenvalue varieties of its torus splittings. Combining general results on eigenvalue varieties of abelian trees of groups with combinatorial descriptions of graph link-manifolds, we construct top-dimensional components in their eigenvalue varieties.
