Automorphisms of free groups and Outer spacepi.math.cornell.edu/~vogtmann/papers/KleinSix.pdfcomplex based on ﬁnite graphs with additional structure. In the Lie case this chain complex

Automorphisms of free groups and Outer space

Lecture 6

Thursday, June 10, 2010

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order to make On finite-dimensional, we also as-sume that the graphs X are connected and haveno vertices of valence one or two. Finally, it isusually convenient to normalize by assuming thesum of the lengths of the edges is equal to one.

The mapping class group acts on Teichmullerspace by changing the marking, and the anal-ogous statement is true here: an element ofOut(Fn), represented by a homotopy equivalenceof R, acts on Outer space by changing only themarking, not the metric graph. A major differ-ence from Teichmuller space appears when onelooks closely at a neighborhood of a point. Te-ichmuller space is a manifold. In Outer spacepoints arbitrarily close to a given point (X, g)may be of the form (Y, h) with Y not homeo-morphic to X. An example is shown in Figure 1,

where several marked graphs near the red graphare obtained by folding pairs of edges incidentto the vertex x together for a small distance. Ingeneral there are many different possible foldings,and this translates to the fact that there is noEuclidean coordinate system which describes allnearby points, i.e. Outer space is not a manifold.

Outer space is not too wild, however...it doeshave the structure of a locally finite cell com-plex, and it is a theorem that Outer space is con-tractible. It also has the structure of a unionof open simplices, each obtained by varying theedge-lengths of a given marked graph (X, g). Forn = 2 these simplices can have dimension 1 or 2,but not dimension 0, so Outer space is a union ofopen triangles identified along open edges (Figure2).

Figure 2: Outer space in rank 2

The stabilizer of a point (X, g) under the ac-tion of Out(Fn) is isomorphic to the group ofisometries of the graph X. In particular, it isa finite group, so the action is proper. ThereforeOuter space serves as an appropriate analog ofthe homogeneous space used to study a latticein a semisimple Lie group, or of the Teichmullerspace used to study the mapping class group of asurface.

The analogies with lattices and with map-ping class groups have turned out to be quitestrong. For example, it has been shown thatOut(Fn) shares many cohomological properties,basic subgroup structure and many rigidity prop-erties with these classes of groups. The proofs of

these facts are frequently inspired by proofs in theanalogous settings and use the action of Out(Fn)on Outer space. However, the details are often ofa completely different nature and can vary dra-matically in difficulty, occasionally being easierfor Out(Fn) but more often easier in at least oneof the other settings.

Perhaps the most extensive use of Outer spaceto date has been for computing algebraic invari-ants of Out(Fn) such as cohomology and Eu-ler characteristic. Appropriate variations, sub-spaces, quotient spaces and completions of Outerspace are also used. For example, the fact thatOut(Fn) acts with finite stabilizers on On impliesthat a finite-index, torsion-free subgroup Γ acts

Outer space (n=2)


xy-1xyx-1 y

x y-1

x y

Reduced Outer space(n=2)


Moving inOuter space (n=2)


Closure of Reduced Outer spacein ℙC(n=2)


Closure of Outer spacein ℙC(n=2)


Some of the things I haven’t talked about yet: •Pointed graphs and degree subcomplexes•Homology stability and stable homology•Feynman diagrams and Kontsevich’s theorem•Morita classes and unstable homology•Complexes of free factors, free factorizations, etc.•Rigidity properties of Out(Fn)•Local structure of Outer space and phylogenetic trees•Outer space and tropical geometry•Currents and laminations•The Lipschitz metric on Outer space •Growth rates of automorphisms•Scott conjecture, bounded cancellation and train tracks


Pointed graphs and degree subcomplexes

Spaces on the (bordification) boundary of Outer space Degree subcomplexes of Auter spaceLow-dimensional homology calculationsRational homology stability



Recall: Bestvina and Feighn added cells “at infinity” to obtain the bordification Ôn of Outer space.



Sphere system picture: A simplex of the sphere complex is “at infinity” if it corresponds to a sphere system S in M=#n(S1 x S2) which is not simple.

Cells added in the bordification correspond to sphere systems in the manifold cut open along S, and their degenerations.This manifold MS is #i(S1 x S2) - ∪B3 for some i<n.π0(Diff(MS))/(Dehn twists) acts on this sphere system complex.

π



If we remove only one ball, π0(Diff(MS))/(Dehn twists) is isomorphic to Aut(Fn).

π0(Diff(MS))/(Dehn twists) acts on this sphere system complex.

This manifold MS is #i(S1 x S2) - ∪B3 for some i<n.



Define An,k = subspace of An spanned by graphs of degree ≤ k

Removing one ball from M corresponds to adding a basepoint to graphs; the resulting space is called Auter space An, with an action of Aut(Fn).

Define the degree of a graph to be the number of vertices other than the basepoint (counted with multiplicity).

Degree 1 Degree 2



Therefore, can use An,k /Aut(Fn) to compute Hk-1(Aut(Fn);ℚ)

Theorem [Hatcher-V]: An,k is k dimensional and (k-1)-connected.

A4,2 /Aut(F3)

Results: Hi(Aut(Fn); ℚ)=0 i ≤ 7 except H4(Aut(F4); ℚ) = ℚ.

Note: The class in H4(Aut(F4); ℚ) survives in H4(Out(F4); ℚ).

Define An,k = subspace of An spanned by graphs of degree ≤ k

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Theorem [Hatcher-V]: Hi(Aut(Fn)) → Hi(Aut(Fn+1)) is an isomorphism for n>>i.

Simple proof for rational homology: An,k /Aut(Fn) is homeomorphic to

An+1,k /Aut(Fn+1) for n > 3k/2.

There are natural inclusions Aut(Fn) ↪ Aut(Fn+1)

So the rational homology is the same in this range.

(With more work can improve the range and prove integral homology stability.)

A4,2 /Aut(F3)

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Questions:

• Is there a degree theorem for Outer space?

• What is the exact rational stability range?

• Is there any unstable homology above dimension 4?

• How long does the unstable homology persist under the stabilization map?



Homology stability and stable homology

Tethered spheres and integral homology stability

To prove homology stability, want a highly-connected Aut(Fn)- complex where the stabilizer of a vertex is Aut(Fn-1).

The action of Aut(Fn) on this complex gives a spectral sequence where one of the maps is the stabilization map Hi(Aut(Fn-1))→ Hi(Aut(Fn))

A straightforward argument then gives you homology stability.


Homology stability and sphere systems

M= #n(S1 x S2) - B3

Look at co-connected sphere systems S0(M).

Theorem [Hatcher] S0(M) is (n-3)-connected.



Aut(Fn) acts on S0(M).

In S0(M) the stabilizer of a single sphere (i.e. a vertex) is π0Diff(M,B1,B2,B3)/Dehn twists.

Theorem [Hatcher] S0(M) is (n-3)-connected.



Stabilizer of a vertex is now Aut(Fn-1), and we have

Theorem [Hatcher-V]: The complex T(M) of tethered co-connected sphere systems is (n-3)/2-connected.

Proof of homology stability is now straightforward

To fix this: connect new boundary spheres to base sphere by “tethers”.


Homology stability and stable homology

Tethered spheres and integral homology stabilityHomology of the map Aut(Fn) to Out(Fn)Stable homology and homotopy theory (Galatius theorem)

The tethered sphere complex gives a new proof of integral homology stability for Aut(Fn). In addition, the map from Aut(Fn) to Out(Fn) was shown to be an isomorphism on homology for n sufficiently large, giving “stability” for the homology of Out(Fn). (This statement needs the erratum of Hatcher-V-Wahl). Out(Fn) contains a copy of the symmetric group Sn, which permutes the generators. Galatius proved that the inclusion of Sn in Out(Fn) is an isomorphism on stable homology, i.e. that the stable homology of Out(Fn) is the homology of the sphere spectrum Ω∞S∞. The method of proof he developed also gives a simpler way to prove the Madsen-Weiss theorem computing the stable homology of mapping class groups.


Feynman diagrams and Kontsevich’s theorem

Cyclic operads and graph homologyKontsevich’s theoremA different approach to computing cohomology of Out(Fn) arises from work of Kontsevich related to Feynman diagrams. (These are graphs that represent the trajectories of particles in intermediate stages of a scattering process and which are used to compute Feynman integrals.) Kontsevich considered particular subalgebras of the Lie algebra of derivations of free commutative, associative and Lie algebras, namely those which kill a certain “symplectic element” in each. He used invariant theory to show that in each case the cohomology of these Lie algebras can be identified with the homology of a chain complex based on finite graphs with additional structure. In the Lie case this chain complex of graphs computes the homology of the moduli space of graphs, i.e. the homology of Out(Fn). The associative case computes the homology of the moduli space of punctured surfaces, and the commutative case gives what is called Kontsevich’s graph homology. Graph homology contains invariants of homology spheres.The Lie, associative and commutative operads are examples of cyclic operads, and Kontsevich’s construction makes sense for any cyclic operad, giving new flavors of “graph homology”.


Morita classes and unstable homology

Morita’s constructionDegree of instabilityMorita used Kontsevich’s theorem to construct cocycles in the cohomology of Out(Fn). The idea is to try to compute the abelianization of Kontsevich’s Lie algebra l∞. In an abelian Lie algebra all the differentials in the chain complex computing cohomology are zero, so any non-trivial element of the abelianization gives a cohomology class. Morita found a series of classes in this abelianization, then pulled them back to the cohomology of l∞.

By Kontsevich’s theorem, these correspond to cycles in H4k(Out(F2k+2);ℚ). Morita showed that the first is a non-trivial homology class, so corresponds to the class computed by Hatcher-V. Conant-V showed that the second class is non-trivial. There is no general proof that they are all non-trivial. These classes all lift to Aut(Fn), and Conant-V showed that they all vanish after a single application of the stabilization map Aut(Fn) to Aut(Fn+1).Recent developments: (1) A computer calculation seems to show that the third class is non-trivial. (2) Kassabov has found more pieces of the abelianization of l∞., giving more potential homology classes. These classes are related to modular forms.


Morita classes and unstable homology

Morita’s constructionDegree of instability

Questions: •Are the Morita classes all non-trivial?•Does Kassabov’s construction give the entire abelianization of l∞ ?•Do Kassabov’s classes survive one or more stabilizations?•Do any of these classes map non-trivially to the homology of GL(n,Z)?•Is there any other unstable homology?


Complexes of free factors, free factorizations, etc.Complexes of free factorizations and free factorsSolomon-Tits theoremsThe curve complex of a surface has proved an interesting and useful object. It can be thought of as living “at infinity” of Teichmuller space...it can be embedded in the Thurston compactification, and it parametrizes the Harvey bordification. The fact that it is δ-hyperbolic is useful for resolving rigidity questions, among many other things.One would like a similar complex for Out(Fn). One natural candidate is the complex of free factorizations, since a single free splitting of Fn corresponds to a single sphere in M, which sits at infinity in S(M); in fact the entire 1-skeleton of S(M) lives at infinity. The description of points on the bordification in terms of chains G ⊃G1⊃G2⊃...⊃Gk suggests considering the complex of free factors, instead of free factorizations. This gives a stronger analogy with the SL(n, ℤ) picture, where the Borel-Serre bordification is parametrized by the spherical Tits building, whose vertices are flags of subspaces of ℝn. There are other variations on this idea, e.g. the complex of cyclic splittings (simplicial trees with edge stabilizers isomorphic to ℤ).Hatcher-V proved that both of these complexes are homotopy equivalent to wedges of spheres, strengthening the analogy with the curve complex (and useful for studying the homology of Out(Fn).


Complexes of free factors, free factorizations, etc.Complexes of free factorizations and free factorsSolomon-Tits theorems

Question: Are any of these complexes δ-hyperbolic?

Kapovich-Lustig have shown that they all have infinite diameter.Bestvina-Feighn have produced δ-hyperbolic complexes on which Out(Fn) acts. However, there is a different complex for each finite set of automorphisms, and other automorphisms do not act nicely on this complex; in particular, infinite order irreducible automorphisms may have fixed points. Sabalka and Savchuk have produced flats in the free factorization complex, showing that it is not δ-hyperbolic.


Rigidity properties of Out(Fn)Maps from lattices to Out(Fn)Maps from Out(Fn)Maps from Out(Fn) to Out(Fm)Actions of Out(Fn) on spheres and contractible manifoldsBridson-Wade have shown that any map of an irreducible lattice in a higher-rank semisimple Lie group to Out(Fn) has finite image. This relies on work of many people, including Magnus, Bestvina-Feighn, Handel-Mosher, Hammenstadt,...Bridson-V showed that if G does not contain the symmetric group Sn+1, then any map from Out(Fn) to G has finite image. Khramtsov showed that Out(Fn) does not embed in Out(Fn+1), but that Out(F2) embeds in Out(F4). Bridson-V showed that any map Out(Fn) to Out(Fm) has finite image if m≤2n, m≠n. Bogopolski-Puga showed that Out(Fn) embeds in Out(Fm) for certain large values of m. Out(Fn) maps to GL(n,ℤ), so acts on ℝn and on its boundary sphere Sn-1. Bridson-V showed that any action by homeomorphisms on a Euclidean space or sphere of smaller dimension must be trivial, i.e. fix every point.Bestvina-Kapovich-Kleiner show that Out(Fn) cannot act properly discotinuously on any manifold of dimension < 4n-6.


Questions:

• What about maps from finite index subgroups of Out(Fn)?• What about actions of finite index subgroups?• Can Out(Fn) act faithfully on a sphere?

Rigidity properties of Out(Fn)


Currents and laminations

Laminations

Projective currents

Closure of Outer space in the space of projective currents

Intersection pairing between currents and trees


The Lipschitz metric on Outer space

The minimal stretch metric

Folding paths as geodesics

Axes for fully irreducible automorphisms

Algom-Kfir’s theorem


Growth rates

Perron-Frobenius eigenvalues as growth rates of automorphisms. (So growth rates are algebraic integers)

Growth rates of f and f-1

Thurston example: growth rates which are not algebraic units.

Realizing various types of algebraic numbers (e.g. Pissot numbers)


Local structure of the spine of Outer space

Cohen-Macaulay property

Symmetries of the spine

Phylogenetic trees


Outer space and tropical geometry

Upper links in the spine and (phylogenetic) trees

Phylogenetic trees and the tropical grassmannian

The “moduli space of tropical curves”

The Jacobian map, failure of the Torelli theorem, the Schottky problem


Scott conjecture, bounded cancellation and train tracks

Scott’s theorem for finite-order automorphisms

Bounded cancellation and the Scott conjecture

Train tracks for fully irreducible automorphisms

Arboreal proof of the Scott conjecture, using train tracks for fully irreducible automorphisms


New Outer spaces

Guirardel-Levitt deformation spaces

Outer spaces for right-angled Artin groups



Documents

Automorphisms of free groups and Outer spacepi.math.cornell.edu/~vogtmann/papers/KleinSix.pdfcomplex based on ﬁnite graphs with additional structure. In the Lie case this chain complex