Out ( F n ) Outer space and the work of Karen Vogtmann F n = < a - PDF document

Out ( F n ) Outer space and the work of Karen Vogtmann F n = < a 1 , a 2 , · · · , a n > is the free group of rank n . Out ( F n ) = Aut ( F n ) / Inn ( F n ) ◮ contains MCG ( S ) for punctured surfaces S ◮ maps to GL n ( Z ) The study of mapping class groups and arithmetic groups is an inspiration in the study of Out ( F n ). Theorem (Nielsen, 1924) Aut ( F n ) (and Out ( F n ) ) are finitely presented. A generating set consists of the automorphisms σ : a 1 �→ a 1 a 2 , a i �→ a i for i > 1 plus the signed permutations of the a i ’s. Outer space Outer space Definition Definition (Culler-Vogtmann, 1986) ◮ graph: finite 1-dimensional cell complex Γ, all vertices have Outer space X n is the space of equivalence classes of marked valence ≥ 3. metric graphs ( g , Γ) where ( g , Γ) ∼ ( g ′ , Γ ′ ) if there is an isometry ◮ rose R = R n : wedge of n circles. φ : Γ → Γ ′ so that g ′ φ ≃ g . a b aba Γ ab g b a ց aB c φ ↓ R b ր g ′ ◮ marking: homotopy equivalence g : Γ → R . Γ ′ ◮ metric on Γ: assignment of positive lengths to the edges of Γ so that the sum is 1. Outer space in rank 2 aB b a b B a Triangles have to be added to edges along the base.

Outer space Topology (3 approaches, all equivalent): ◮ simplicial, with respect to the obvious decomposition into “simplices with missing faces”. ◮ ( g , Γ) is close to ( g ′ , Γ ′ ) if there is a (1 + ǫ )-Lipschitz map f : Γ → Γ ′ with g ′ f ≃ g . ◮ via length functions: if α is a conjugacy class in F n let ℓ ( g , Γ) ( α ) be the length in Γ of the unique immersed curve a such that g ( a ) represents α . Then, for S =set of conjugacy classes X n → [0 , ∞ ) S ( g , Γ) �→ ( α �→ ℓ ( g , Γ) ( α )) is injective – take the induced topology. Theorem (Culler-Vogtmann, 1986) X n is contractible. Action Topological properties If φ ∈ Out ( F n ) let f : R → R be a h.e. with π 1 ( f ) = φ and define ◮ Virtually finite K ( G , 1) (Culler-Vogtmann 1986). g f ◮ vcd ( Out ( F n )) = 2 n − 3 ( n ≥ 2) (Culler-Vogtmann 1986). φ ( g , Γ) = ( fg , Γ) Γ → R n → R n ◮ every finite subgroup fixes a point of X n . ◮ action is simplicial, ◮ every solvable subgroup is finitely generated and virtually ◮ point stabilizers are finite. abelian (Alibegovi´ c 2002) ◮ there are finitely many orbits of simplices (but the quotient is ◮ Tits alternative: every subgroup H ⊂ Out ( F n ) either contains not compact). a free group or is virtually abelian (B-Feighn-Handel, 2000, ◮ the action is cocompact on the spine SX n ⊂ X n . 2005) ◮ Bieri-Eckmann duality (B-Feighn 2000) H i ( G ; M ) ∼ = H d − i ( G ; M ⊗ D ) ◮ Homological stability (Hatcher 1995, Hatcher-Vogtmann 2004) H i ( Aut ( F n )) ∼ = H i ( Aut ( F n +1 )) for n >> i ◮ Computation of stable homology (Galatius, to appear) Dictionary 1 Links SL 2 ( Z ) SL n ( Z ) MCG ( S ) Out ( F n ) An n -complex is Cohen-Macauley if the link of every k -cell is homotopy equivalent to a wedge of ( n − k − 1)-spheres (for every Trace Jordan normal Nielsen- train-tracks k ). form Thurston theory Examples H 2 symmetric Teichm¨ uller Outer space ◮ manifolds, space space ◮ buildings. hyperbolic semi-simple pseudo-Anosov fully irreducible (Anosov) (diagonaliz- mapping class automorphism element able) Theorem (Vogtmann 1990) shear parabolic Dehn twist polynomially Both Outer space and its spine are Cohen-Macauley. growing auto- morphism

Homological stability Homological stability Some groups come in natural sequences G 1 ⊂ G 2 ⊂ G 3 ⊂ · · · Examples ◮ permutation groups S n , Theorem ◮ signed permutation groups S ± n , Aut ( F n ) satisfies homological stability. ◮ braid groups, Hatcher 1995, Hatcher-Vogtmann 1998, Hatcher-Vogtmann 2004, ◮ SL n ( Z ), O n , n ( Z ), Hatcher-Vogtmann-Wahl 2006. The proof over Q is particularly striking, from [Hatcher-Vogtmann ◮ mapping class groups of surfaces with one boundary 1998]. component, ◮ Aut ( F n ). The sequence satisfies homological stability if for every k for sufficiently large n ∼ = H k ( G n ) → H k ( G n +1 ) is an isomorphism. Homological stability Homological stability We are interested in H k − 1 ( Aut ( F n ) , Q ) = H k − 1 ( A n , k / Aut ( F n ) , Q ). Aut ( F n ) acts on Autre espace: this is the space of marked metric Homological stability follows from: graphs with a basepoint. The degree of a graph is 2 n − (valence at the basepoint). A n , k / Aut ( F n ) = A n +1 , k / Aut ( F n +1 ) for n ≥ 2 k . 1 0 A n , k / Aut ( F n ) is the space of unmarked metric graphs. Increasing n amounts to wedging a loop at the basepoint. When n 2 2 is large every degree k graph has Degree Theorem. The space A n , k of graphs in the spine of a loop at the basepoint. degree ≤ k is ( k − 1)-connected. [H-V 1998], [Bux-McEwen 2009] Homological stability Dynamical properties ◮ X n can be equivariantly compactified to X n (Culler-Morgan, One can give a proof over Z along the same lines. 1987), analogous to Thurston’s compactification of Stability holds for Teichm¨ uller space via projective measured laminations. ◮ symmetric groups S n (Nakaoka 1960, Maazen 1979 simpler ◮ proof), X n ⊂ [0 , ∞ ) S → P [0 , ∞ ) S ◮ signed symmetric groups S ± n (Maazen’s proof easily modifies.) is injective; take the closure. ◮ H × S ± n for a fixed group H (K¨ unneth formula). View A n , k / Aut ( F n ) as an “orbihedron”; pay attention to the (finite) stabilizers. E.g. at the rose, the stabilizers are signed permutation groups S ± n = S n ⋊ Z n 2 . Stability holds for this sequence. In fact, stability holds for the sequence of stabilizers at every point of the orbihedron. They all have the form H × S ± n ◮ A point of X n can be viewed as a free simplicial F n -tree; a for a fixed group H . point in ∂ X n = X n \ X n is an F n -tree (not necessarily free nor simplicial).

Dynamical properties Dictionary 2 MCG ( S ) Out ( F n ) simple closed curve primitive conjugacy class ◮ Points in ∂ X n can be studied using the Rips machine. free factor incompressible subsurface splitting of F n ◮ Guirardel (2000): action on ∂ X n does not have dense orbits. measured lamination R -tree He also conjecturally identified the minimal closed invariant Thurston’s boundary Culler-Morgan’s boundary set. attracting lamination for a attracting tree for a fully irre- ◮ North-South dynamics for fully irreducible elements pseudo-Anosov ducible automorphism (Levitt-Lustig, 2003) measured geodesic current measured geodesic current intersection number between length of a current in an R -tree measured laminations free factor complex curve complex splitting complex Marc Culler and Karen Vogtmann. Moduli of graphs and automorphisms of free groups. Invent. Math. , 84(1):91–119, 1986. Karen Vogtmann. Automorphisms of free groups and outer space. In Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000) , volume 94, pages 1–31, 2002. Karen Vogtmann. The cohomology of automorphism groups of free groups. In International Congress of Mathematicians. Vol. II , pages 1101–1117. Eur. Math. Soc., Z¨ urich, 2006. Martin R. Bridson and Karen Vogtmann. Automorphism groups of free groups, surface groups and free abelian groups. In Problems on mapping class groups and related topics , volume 74 of Proc. Sympos. Pure Math. , pages 301–316. Amer. Math. Soc., Providence, RI, 2006.