# The geometry of the outer automorphism group of a free group

October 25 to October 29, 2010

at the

American Institute of Mathematics, San Jose, California

organized by

Matt Clay, Vincent Guirardel, and Alexandra Pettet

## Original Announcement

This workshop 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.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

The Dehn functions of $Out(F_n)$ and $Aut(F_n)$
by  Martin R. Bridson and Karen Vogtmann,  Ann. Inst. Fourier (Grenoble) 62 (2012), no. 5, 1811-1817  MR3025154
Hyperbolicity of the complex of free factors
by  Mladen Bestvina and Mark Feighn,  Adv. Math. 256 (2014), 104-155  MR3177291
Indecomposable $F_N$-trees and minimal laminations
by  Thierry Coulbois, Arnaud Hilion, and Patrick Reynolds,  Groups Geom. Dyn. 9 (2015), no. 2, 567-597  MR3356976