International Congress of Mathematicians

ICM 2002 • Vol. II • 373-384The Topology of Out(F n )Mladen Bestvina*AbstractWe will survey the work on the topology of Out(F n ) in the last 20 yearsor so. Much of the development is driven by the tantalizing analogy withmapping class groups. Unfortunately, Out(F n ) is more complicated and lesswell-behaved.Culler and Vogtmann constructed Outer Space X n , the analog of Teichmüllerspace, a contractible complex on which Out(F n ) acts with finitestabilizers. Paths in X n can be generated using "foldings" of graphs, an operationintroduced by Stallings to give alternative solutions for many algorithmicquestions about free groups. The most conceptual proof of the contractibilityof X n involves folding.There is a normal form of an automorphism, analogous to Thurston's normalform for surface homeomorphisms. This normal form, called a "(relative)train track map", consists of a cellular map on a graph and has good propertieswith respect to iteration. One may think of building an automorphismin stages, adding to the previous stages a building block that either growsexponentially or polynomially. A complicating feature is that these blocks arenot "disjoint" as in Thurston's theory, but interact as upper stages can mapover the lower stages.Applications include the study of growth rates (a surprising feature of freegroup automorphisms is that the growth rate of / is generally different fromthe growth rate of / _1 ), of the fixed subgroup of a given automorphism, andthe proof of the Tits alternative for Out(F n ). For the latter, in addition totrain track methods, one needs to consider an appropriate version of "attractinglaminations" to understand the dynamics of exponentially growingautomorphisms and run the "ping-pong" argument. The Tits alternative isthus reduced to groups consisting of polynomially growing automorphisms,and this is handled by the analog of Kolchin's theorem (this is one instancewhere Out(F n ) resembles GL„(Z) more than a mapping class group).Morse theory has made its appearance in the subject in several guises.The original proof of the contractibility of X n used a kind of "combinatorial"Morse function (adding contractible subcomplexes one at a time and studyingthe intersections). Hatcher-Vogtmann developed a "Cerf theory" for graphs.This is a parametrized version of Morse theory and it allows them to provehomological stability results. One can "bordify" Outer Space (by analogywith the Borei-Serre construction for arithmetic groups) to make the action'Department of Mathematics, University of Utah, USA. E-mail: bestvina@math.utah.edu