The geometry of the outer automorphism group of a free group

This workshop, sponsored by AIM and the NSF, will be devoted to the outer automorphism group of the free group, Out(F); in particular, its geometry and its inherent asymmetry.

Many open problems concerning Out(F) are motivated by its connections with arithmetic groups and mapping class groups. As an analog of symmetric spaces and Teichmuller spaces, Outer space is the premier example in a growing dictionary between these groups. It is thus striking how remarkably few of the metric properties of Outer space have been explored; this is in sharp contrast with the classically understood non-positively curved metrics on symmetric spaces, or the well-known geometries on Teichmuller space, such as the Thurston (Lipschitz), Teichmuller, or Weil-Petersson metrics. Furthermore, while the hyperbolicity of the complex of curves has become an indispensable tool for studying the mapping class groups, there is no such technology yet available for Out(F).

The following list of topics will be discussed at the workshop All participants are invited to submit problems or additional topics for discussion during the workshop.

• There are several recent results about the outer automorphism group which lack a uniformity that is present for their mapping class group analog. We hope to determine whether these statements have a uniform version for Out(F).
• Very little is known about the geometry of Outer space. Recently, Algom-Kfir showed that axes of fully irreducible automorphisms satisfy a property associated to negatively curved spaces. Geodesics that are not axes of fully irreducible automorphisms are a mystery. We will try to develop further a theory of geodesics in Outer space with the asymmetric metric; in particular, we will investigate to what extent the work of Handel-Mosher and Algom-Kfir can be extended to geodesics that are not the axis of a fully irreducible automorphism.
• Bestvina-Feighn's hyperbolic complexes are the closest known analog of the complex of curves for mapping class groups. Unfortunately, they rely on several unnatural choices. There are several more natural candidates to consider, e.g. the free factor complex and the splitting complex; however their geometric properties are still unknown. We will study how these complexes are related; in particular we will look for some coarse geometric relations.
• One of the useful features of the complex constructed by Bestvina-Feighn is that it satisfies a weak notion of proper discontinuity. We shall work toward determining whether this weak form of proper discontinuity holds for the free factor complex or the splitting complex.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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