ICM 2002 · Vol. III · 1–3

The topology of Out(F_{n})

Mladen Bestvina^{∗}

2000 Mathematics Subject Classification: 57M07, 20F65, 20E08 Keywords and Phrases: Free group, train tracks, Outer space

1. Introduction

The aim of this note is to survey some of the topological methods developed in the last 20 years to study the group Out(F_{n}) of outer automorphisms of a free group F_{n} of rank n. For an excellent and more detailed survey see also [69]. Stallings’ paper [64] marks the turning point and for the earlier history of the subject the reader is referred to [55]. Out(F_{n}) is defined as the quotient of the group Aut(F_{n}) of all automorphisms of F_{n} by the subgroup of inner automorphisms. On one hand, abelianizing F_{n} produces an epimorphism Out(F_{n}) → Out(ℤ^{n}) = GL_{n}(ℤ), and on the other hand Out(F_{n}) contains as a subgroup the mapping class group of any compact surface with fundamental group F_{n}. A leitmotiv in the subject, promoted by Karen Vogtmann, is that Out(F_{n}) satisfies a mix of properties, some inherited from mapping class groups, and others from arithmetic groups. The table below summarizes the parallels between topological objects associated with these groups.

Mapping  Out(F_{n})  GL_{n}(ℤ)  algebraic
class groups  (arithmetic groups) properties
Teichmüller  Culler-Vogtmann’s GL_{n}(R)/O_{n}  finiteness properties
space  Outer space  (symmetric spaces)  cohomological dimension
Thurston  train track  Jordan  growth rates
normal form  representative  normal form  fixed points (subgroups)
Harer’s  bordification of  Borel-Serre  Bieri-Eckmann
bordification  Outer space  bordification  duality
measured  R-trees  flag manifold  Kolchin theorem
laminations  (Furstenberg boundary) Tits alternative
Harvey’s ? Tits rigidity
curve complex  building

Outer space is not a manifold and only a polyhedron, imposing a combinatorial character on Out(F_{n}).

∗Department of Mathematics, University of Utah, USA. E-mail: bestvina@math.utah.edu

2. Stallings’ Folds

A graph is a 1-dimensional cell complex. A map f : G → G′ between graphs is simplicial if it maps vertices to vertices and open 1-cells homeomorphically to open 1-cells. The simplicial map f is a fold if it is surjective and identifies two edges that share at least one vertex. A fold is a homotopy equivalence unless the two edges share both pairs of endpoints and in that case the induced homomorphism in π_{1} corresponds to killing a basis element.

Theorem 1 (Stallings [63]). A simplicial map f : G → G′ between finite connected graphs can be factored as the composition

G = G_{0} → G_{1} → ⋯ → G_{k} → G′

where each G_{i} → G_{i+1} is a fold and G_{k} → 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 G_{k} → G′ can be thought of as the core of the covering space of G′ corresponding to the image in π_{1} of f. The following problems can be solved algorithmically using Theorem 1 (these were known earlier, but Theorem 1 provides a simple unified argument). Let F be a free group with a fixed finite basis.

• Find a basis of the subgroup H generated by a given finite collection h_{1}, ⋯, h_{k} of elements of F.

• Given w ∈ F, decide if w ∈< h_{1}, ⋯, h_{k} >.

• Given w ∈ F, decide if w is conjugate into < h_{1}, ⋯, h_{k} >.

• Given a homomorphism ϕ : F → F′ between two free groups of finite rank, decide if ϕ is injective, surjective.

• Given finitely generated H < F decide if it has finite index.

• Given two f.g. subgroups H_{1}, H_{2} < F compute H_{1} ∩ H_{2} and also the collection of subgroups H_{1} ∩ H_{2}^{g} where g ∈ F. In particular, is H_{1} malnormal?

• Represent a given automorphism of F as the composition of generators of Aut(F)

of the following form:

Signed permutations: each a_{i} maps to a_{i} or to a_{i}^{−1}.

Change of maximal tree: a_{1} ↦ a_{1} and for i > 1 a_{i} maps to one of a_{1}^{±1}a_{i} or to a_{i}a_{1}^{±1}.

• Todd-Coxeter process [65].

3. Culler-Vogtmann’s Outer space

Fix the wedge of n circles R_{n} and a natural identification π_{1}(R_{n}) ≅ F_{n} in which oriented edges correspond to the basis elements. Thus any ϕ ∈ Out(F_{n}) can be thought of as a homotopy equivalence R_{n} → R_{n}. A marked metric graph is a pair (G, g) where

• G is a finite graph without vertices of valence 1 or 2.

• g : R_{n} → G is a homotopy equivalence (the marking).

The topology of Out(F_{n}) 3

• G is equipped with a path metric so that the sum of the lengths of all edges is 1.

Outer space X_{n} is the set of equivalence classes of marked metric graphs under the equivalence relation (G, g) ∼ (G′, g′) if there is an isometry h : G → G′ such that gh and g′ are homotopic [28].

If α is a loop in R_{n} we have the length function l_{α} : X_{n} → ℝ where l_{α}(G, g) is the length of the immersed loop homotopic to g(α). The collection {l_{α}} as α ranges over all immersed loops in R_{n} defines an injection X_{n} → ℝ^{∞} and the topology on X_{n} is defined so that this injection is an embedding. X_{n} naturally decomposes into open simplices obtained by varying edge-lengths on a fixed marked graph. The group Out(F_{n}) acts on X_{n} on the right via

(G, g)ϕ = (G, gϕ)

Theorem 2 (Culler-Vogtmann [28]). X_{n} is contractible and the action of Out(F_{n}) is properly discontinuous (with finite point stabilizers). X_{n} equivariantly deformation retracts to a (2n − 3)-dimensional complex.

If (G, g) and (G′, g′) represent two points of X_{n}, there is a “difference of markings” map h : G → G′ such that hg and g′ are homotopic. Representing h as a composition of folds (appropriately interpreted) leads to a path in X_{n} from (G, g) to (G′, g′). Arranging that these paths vary continuously with endpoints leads to a proof of contractibility of X_{n} [66],[60],[71].

Corollary 3. The virtual cohomological dimension vcd(Out(F_{n})) = 2n − 3.

Theorem 4 (Culler [26]). Every finite subgroup of Out(F_{n}) fixes a point of X_{n}.

Outer space can be equivariantly compactified [27]. Points at infinity are represented by actions of F_{n} on ℝ-trees.

4. Train tracks

Any ϕ ∈ Out(F_{n}) can be represented as a cellular map f : G → G on a marked graph G. We say that ϕ is reducible if there is such a representative where

• G has no vertices of valence 1 or 2, and

• there is a proper f-invariant subgraph of G with at least one non-contractible component.

Otherwise, we say that ϕ is irreducible.

A cellular map f : G → G is a train track map if for every k > 0 the map f^{k} : G → G is locally injective on every open 1-cell. For example, homeomorphisms are train track maps and Culler’s theorem guarantees that every ϕ ∈ Out(F_{n}) of finite order has a representative f : G → G which is a homeomorphism. More generally, we have

Theorem 5 (Bestvina-Handel [12]). Every irreducible outer automorphism ϕ can be represented as a train track map f : G → G.

Any vertex v ∈ G has a cone neighborhood, and the frontier points can be thought of as “germs of edges” at v. A train track map (or any cellular map that does not collapse edges) f induces the “derivative” map Df on these germs (on possibly different vertices). We declare two germs at the same vertex to be equivalent (and the corresponding “turn” illegal) if they get identified by some power of Df (and otherwise the turn is legal). An immersed loop in G is legal if every turn determined by entering and then exiting a vertex is legal. It follows that f sends legal loops to legal loops. This gives a method for computing the growth rate of ϕ, as follows. The transition matrix (a_{ij}) of f (or more generally of a cellular map G → G that is locally injective on edges) has a_{ij} equal to the number of times that the f-image of j^{th} edge crosses i^{th} edge. Applying the Perron-Frobenius theorem to the transition matrix, one can find a unique metric structure on G such that f expands lengths of edges (and also legal loops) by a factor λ ≥ 1. For a conjugacy class γ in F_{n} the growth rate is defined as

GR(ϕ, γ) = lim sup_{k→∞} log(||ϕ^{k}(γ)||)/k

where ||γ|| is the word length of the cyclically reduced word representing γ. Growth rates can be computed using lengths of loops in G rather than in R_{n}.

Corollary 6. If ϕ is irreducible as above, then either γ is a ϕ-periodic conjugacy class, or GR(ϕ, γ) = log λ. Moreover, lim sup can be replaced by lim.

The proof of Theorem 5 uses a folding process that successively reduces the Perron-Frobenius number of the transition matrix until either a train track representative is found, or else a reduction of ϕ is discovered. This process is algorithmic (see [13],[21]).

Another application of train tracks is to fixed subgroups.

Theorem 7 (Bestvina-Handel [12]). Let Φ : F_{n} → F_{n} be an automorphism whose associated outer automorphism is irreducible. Then the fixed subgroup F ix(Φ) is trivial or cyclic. Without the irreducibility assumption, the rank of F ix(Φ) is at most n.

It was known earlier by the work of Gersten [39] that F ix(Φ) has finite rank (for simpler proofs see [42],[25]). The last sentence in the above theorem was conjectured by Peter Scott. Subsequent work by Collins-Turner [24], Dicks-Ventura [31], Ventura [68], Martino-Ventura [58], imposed further restrictions on a subgroup of F_{n} that occurs as the fixed subgroup of an automorphism. To analyze reducible automorphisms, a more general version of a train track map is required.

Definition 8. A cellular map f : G → G on a finite graph with no vertices of valence 1 that does not collapse any edges is a relative train track map if there is a filtration

∅ = G_{0} ⊂ ⋯ ⊂ G_{m} = G

into f-invariant subgraphs with the following properties. Denote by H_{r} the closure of G_{r}\G_{r−1}, and by M_{r} the part of the transition matrix corresponding to H_{r}. Then M_{r} is the zero matrix or an irreducible matrix. If M_{r} is irreducible and the Perron-Frobenius eigenvalue λ_{r} > 1 then:

• the derivative Df maps the germs in H_{r} to germs in H_{r},

• if α is a nontrivial path in G_{r−1} with endpoints in G_{r−1} ∩ H_{r} then f(α), after pulling tight, is also a nontrivial path with endpoints in G_{r−1} ∩ H_{r}, and

• every legal path in H_{r} is mapped to a path that does not cross illegal turns in H_{r}.

As an example, consider the automorphism a ↦ a, b ↦ ab, c ↦ caba^{−1}b^{−1}d, d ↦ dbcd represented on the rose R_{4}. The strata are ∅ ⊂ G_{1} = {a} ⊂ {a, b} ⊂ G. H_{1} and H_{2} have λ = 1 while H_{3} has λ_{3} > 1. The following is an analog of Thurston’s normal form for surface homeomorphisms.

Theorem 9. [12] Every automorphism of F_{n} admits a relative train track representative.

Consequently, automorphisms of F_{n} can be thought of as being built from building blocks (exponential and non-exponential kinds) but the later stages are allowed to map over the previous stages. This makes the study of automorphisms of F_{n} more difficult (and interesting) than the study of surface homeomorphisms. Other nonsurface phenomena (present in linear groups) are:

• stacking up non-exponential strata produces (nonlinear) polynomial growth,

• the growth rate of an automorphism is generally different from the growth rate of its inverse.

5. Related spaces and structures

Unfortunately, relative train track representatives are far from unique. As a replacement, one looks for canonical objects associated to automorphisms that can be computed using relative train tracks. There are 3 kinds of such objects, all stemming from the surface theory: laminations, ℝ-trees, and hierarchical decompositions of F_{n} [59]. Laminations. Laminations were used in the proof of the Tits alternative for Out(F_{n}). To each automorphism one associates finitely many attracting laminations. Each consists of a collection of “leaves”, i.e. biinfinite paths in the graph G, or alternatively, of an F_{n}-orbit of pairs of distinct points in the Cantor set of ends of F_{n}. A leaf ℓ can be computed by iterating an edge in an exponentially growing stratum H_{r}. The other leaves are biinfinite paths whose finite subpaths appear as subpaths of ℓ. Some of the attracting laminations may be sublaminations of other attracting laminations, and one focuses on the maximal (or topmost) laminations. It is possible to identify the basin of attraction for each such lamination. Let H be any subgroup of Out(F_{n}). Some of the time it is possible to find to elements f, g ∈ H that attract each other’s laminations and then the standard ping-pong argument shows that < f, g >≅ F_{2}. Otherwise, there is a finite set of attracting laminations permuted by H, a finite index subgroup H_{0} ⊂ H that fixes each of these laminations and a homomorphism (“stretch factor”) H_{0} → A to a finitely generated abelian group A whose kernel consists entirely of polynomially growing automorphisms. There is an analog of Kolchin’s theorem that says that finitely generated groups of polynomially growing automorphisms can simultaneously be realized as relative train track maps on the same graph (the classical Kolchin theorem says that a group of unipotent matrices can be conjugated to be upper triangular, or equivalently that it fixes a point in the flag manifold). The main step in the proof of the analog of Kolchin’s theorem is to find an appropriate fixed ℝ-tree in the boundary of Outer space. This leads to the Tits alternative for Out(F_{n}):

Theorem 10 (Bestvina-Feighn-Handel [9],[10],[7]). Any subgroup H of Out(F_{n}) either contains F_{2} or is virtually solvable.

A companion theorem [8] (for a simpler proof see [1]) is that solvable subgroups of Out(F_{n}) are virtually abelian. ℝ-trees. Points in the compactified Outer space are represented as F_{n}-actions on ℝtrees. It is then not surprising that the Rips machine [5], which is used to understand individual actions, provides a new tool to be deployed to study Out(F_{n}). Gaboriau, Levitt, and Lustig [37] and Sela [59] find another proof of Theorem 7. Gaboriau and Levitt compute the topological dimension of the boundary of Outer Space [36]. Levitt and Lustig show [51] that automorphisms with irreducible powers have the standard north-south dynamics on the compactified Outer space. Guirardel [43] shows that the action of Out(F_{n}) on the boundary does not have dense orbits; however, there is a unique minimal closed invariant set. For other applications of ℝ-trees in geometric group theory, the reader is referred to the survey [2]. Cerf theory. An advantage of Aut(F_{n}) over Out(F_{n}) is that there is a natural inclusion Aut(F_{n}) → Aut(F_{n+1}). One can define Auter Space AX_{n} similarly to Outer space, except that all graphs are equipped with a base vertex, which is allowed to have valence 2. The degree of the base vertex v is 2n − valence(v). Denote by D_{n}^{k} the subcomplex of AX_{n} consisting of graphs of degree ≤ k. Hatcher-Vogtmann [47] develop a version of Cerf theory and show that D_{n}^{k} is (k − 1)-connected. Since the quotient D_{n}^{k}/Aut(F_{n}) stabilizes when n is large, one sees that (rational) homology H_{i}(Aut(F_{n})) also stabilizes when n is large (n ≥ 3i/2). Hatcher-Vogtmann show that the same is true for integral homology and in the range n ≥ 2i + 3. They also make explicit computations in low dimensions [49] and all stable rational homology groups H_{i} vanish for i ≤ 7. Bordification. The action of Out(F_{n}) on Outer space X_{n} is not cocompact. By analogy with Borel-Serre bordification of symmetric spaces [14] and Harer’s bordification of Teichmüller space [44], Bestvina and Feighn [6] bordify X_{n}, i.e. equivariantly add ideal points so that the action on the new space BX_{n} is cocompact. This is done by separately compactifying every simplex with missing faces in X_{n} and then gluing these together. To see the idea, consider the case of the theta-graph in rank 2. Varying metrics yields a 2-simplex σ without the vertices. As a sequence of metrics approaches a missing vertex, the lengths of two edges converge to 0. Restricting a metric to these two edges and normalizing so that the total length is 1 gives a point in [0, 1] (the length of one of the edges), and a way to compactify σ by adding an interval for each missing vertex. The compactified σ is a hexagon. This procedure equips the limiting theta graph with a metric that may vanish on two edges, in which case a “secondary metric” is defined on their union. In general, a graph representing a point in the bordification is equipped with a sequence of metrics, each defined on the core of the subgraph where the previous metric vanishes.

Lengths of curves (at various scales) provide a “Morse function” on BX_{n} with values in a product of [0, ∞)’s with the target lexicographically ordered. The sublevel and superlevel sets intersect each cell in a semi-algebraic set and it is possible to study how the homotopy types change as the level changes. A distinct advantage of BX_{n} over the spine of X_{n} (an equivariant deformation retract) is that the change in homotopy type of superlevel sets as the level decreases is very simple – via attaching of cells of a fixed dimension.

Theorem 11 (Bestvina-Feighn [6]). BX_{n} and Out(F_{n}) are (2n − 5)-connected at infinity, and Out(F_{n}) is a virtual duality group of dimension 2n − 3.

Mapping tori. If ϕ : F_{n} → F_{n} is an automorphism, form the mapping torus M(ϕ). This is the fundamental group of the mapping torus G × [0, 1]/(x, 1) ∼ (f(x), 0) of any representative f : G → G, and it plays the role analogous to 3-manifolds that fiber over the circle. Such a group is always coherent [33]. A quasi-isometry classification of these groups seems out of reach, but the following is known. When ϕ has no periodic conjugacy classes, M(ϕ) is a hyperbolic group [20]. When ϕ has polynomial growth, M(ϕ) satisfies quadratic isoperimetric inequality [57] and moreover, M(ϕ) quasi-isometric to M(ψ) for ψ growing polynomially forces ψ to grow as a polynomial of the same degree [56]. Bridson and Groves announced [16] that M(ϕ) satisfies quadratic isoperimetric inequality for all ϕ. Geometry. Perhaps the biggest challenge in the field is to find a good geometry that goes with Out(F_{n}). The payoff would most likely include rigidity theorems for Out(F_{n}). Both mapping class groups and arithmetic groups act isometrically on spaces of nonpositive curvature. Unfortunately, the results to date for Out(F_{n}) are negative. Bridson [15] showed that Outer space does not admit an equivariant piecewise Euclidean CAT (0) metric. Out(F_{n}) (n > 2) is far from being CAT (0) [17],[40].

An example of a likely rigidity theorem is that higher rank lattices in simple Lie groups do not embed into Out(F_{n}). A possible strategy is to follow the proof in [11] of the analogous fact for mapping class groups. The major missing piece of the puzzle is the replacement for Harvey’s curve complex; a possible candidate is described in [48].

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