# Past Topology Seminar

I will explain some recent work using minimal surfaces to address problems in 3-manifold topology. Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface. If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting. I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three. The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.

We will discuss the concept of $\ell^2$-stability of a group and show some of its rigidity consequences. We provide moreover some very concrete examples of lattices in product of trees that have many interesting properties, $\ell^2$-stability being only one of them.

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. We will prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we will exhibit various examples of Cayley graphs of finitely presented groups (e.g. PGL(5, Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgakopoulos. This is a joint work with Mikael de la Salle.

We study symplectic invariants of the open symplectic manifolds X

obtained by plumbing cotangent bundles of spheres according to a

plumbing tree. We prove that certain models for the Fukaya category F(X)

of closed exact Lagrangians in X and the wrapped Fukaya category W(X)

are related by Koszul duality. As an application, we give explicit

computations of symplectic cohomology essentially for all trees. This is

joint work with Tolga Etg\"u.

Recent developments in 3-dimensional topological quantum field theory allow us to understand the vector spaces assigned to surfaces as spaces of string diagrams. In the Reshetikhin-Turaev model, these string diagrams live inside a handlebody bounding the surface, while in the Turaev-Viro model, they live on the surface itself. There is a "lifting map" from the former to the latter, which sheds new light on a number of constructions. Joint with Gerrit Goosen.

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.

This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.

I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.

This is joint work with Daniel Woodhouse.