International Congress of Mathematicians

The Topology of Out(F n 375MappingOut(F n)GL„(Z)algebraicclass groups(arithmetic groups)propertiesTeichmüllerCuller-Vogtmann'sGL n(R)/O nfiniteness propertiesspaceOuter space(symmetric spaces)cohomological dimensionThurstontrain trackJordangrowth ratesnormal formrepresentativenormal formfixed points (subgroups)Harer'sbordification ofBorei-SerreBieri-EckmannbordificationOuter spacebordificationdualitymeasuredK-treesflag manifoldKolchin theoremlaminations(Furstenberg boundary)Tits alternativeHarvey's?Titsrigiditycurve complexbuilding1-cells. The simplicial map / is a fold if it is surjective and identifies two edges thatshare at least one vertex. A fold is a homotopy equivalence unless the two edgesshare both pairs of endpoints and in that case the induced homomorphism in mcorresponds to killing a basis element.Theorem 1 (Stallings [63]). A simplicial map f : Gnected graphs can be factored as the compositionG' between finite con-G — (jr0 —y (jrx Gk G'where each Gì —¥ Gj+i is a fold and Gk —¥ G' is locally injective (an immersion).Moreover, such a factorization can be found by a (fast) algorithm.In the absence of valence 1 vertices the last map Gk —¥ G' can be thoughtof as the core of the covering space of G' corresponding to the image in m of /.The following problems can be solved algorithmically using Theorem 1 (these wereknown earlier, but Theorem 1 provides a simple unified argument). Let F be a freegroup with a fixed finite basis.• Find a basis of the subgroup H generated by a given finite collection hi,--- ,huof elements of F.• Given w £ F, decide if w £< hi, • • • ,hu >•• Given w £ F, decide if w is conjugate into < hi,- • • ,hu >•• Given a homomorphism : F —t F' between two free groups of finite rank,decide if is injective, surjective.• Given finitely generated H < F decide if it has finite index.• Given two f.g. subgroups Hi,H 2 < F compute HiriH 2 and also the collectionof subgroups Hi fl flf where g £ F. In particular, is Hi malnormal?• Represent a given automorphism of F as the composition of generators ofAut(F) of the following form:Signed permutations: each a, maps to a, or to a,Change of maximal tree: cti H> cti, H> a±i1 a, or a, H> 0,% (i > 1).Todd-Coxeter process [65].