Symplectic derivations and automorphisms of free groups Part II (Karen Vogtmann, University of Warwick)

02.11.2017 10:00 – 11:45

Suppose L(V) is the free Lie algebra on a symplectic vector space V. A theorem of M. Kontsevich neatly relates the cohomology of the Lie algebra $D^\omega(V)$ of symplectic derivations of L(V) with the cohomology of the groups $Out(F_n)$ of outer automorphisms of finitely-generated free groups. The connection is established via finite graphs: on the one hand Weyl’s invariant theory can be used to encode symplectic invariants of the chain complex for $D^\omega(V)$ in terms of graphs, and on the other hand automorphisms of free groups can be identified with homotopy equivalences of graphs. The end result is a concrete combinatorial way of understanding these abstract algebraic invariants.
I will define all of the relevant objects and explain the relation between them, without assuming any prior familiarity from the audience. There are many open problems related to this construction. For example, S. Morita used Kontsevich’s theorem to find a series of cycles for $Out(F_n)$. All of these cycles are conjectured to represent non-trivial homology classes, but this has been proved only for the first three. I will explain how Morita’s cycles can be reinterpreted in terms of "hairy graphs" and then show how this graphical picture leads to the construction of large numbers of new classes, including some based on classical modular forms for $SL(2,Z)$.

Lieu

Bâtiment: Battelle

Séminaire "Groupes de Lie et espaces des modules"

entrée libre