Abstracts for Topics in Topology

We prove that if T is an $\mathbf R$-tree equipped with a minimal free isometric action of $F_N$ then the $Out(F_N)$-stabilizer of the projective class [T] of [T] is virtually cyclic. As an application, we obtain a new proof of the Tits Alternative for "dynamically large" subgroups of $Out(F_N)$. The talk is based on joint work with Martin Lustig.

Discrete dynamical systems given by a continuous map on a topological (usually compact metrizable) space will be considered. Minimality of such a system/map can be defined as the density of all forward orbits. Every compact system contains at least one minimal set, i.e., a nonempty closed invariant subset such that the restriction of the map to this subset is minimal. A fundamental question in topological dynamics is the one on the topological structure of minimal sets of continuous maps in a given space (and, in particular, whether the space itself admits a minimal map or not).

Every finitely presented group is the fundamental group of the total space of a Lefschetz fibration over 2-sphere. This follows from results of Gompf and Donaldson, and was also proved by Amoros-Bogomolov-Katzarkov-Pantev. I will give another proof by providing the monodromy explicitly. Then I will discuss an invariant of finitely presented groups obtained this way.

In 1978, Thurston introduced an affine algebraic set to study hyperbolic structures on triangulated 3-manifolds. Recently, Feng Luo discovered a finite dimensional variational principle on triangulated 3-manifolds with the property that its critical points are related to both Thurston's algebraic set and to Haken's normal surface theory. The action functional is the volume. This is a generalisation of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary.