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WHAT IS Outer Space?

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WHAT IS Outer Space? WHAT IS Outer Space? Karen Vogtmann To investigate the properties of a group G, it is with a metric of constant negative curvature, and often useful to realize G as a group of symmetries g : S ! X is a homeomorphism, called the mark- of some geometric object. For example, the clas- ing, which is well-defined up to isotopy. From this sical modular group P SL(2; Z) can be thought of point of view, the mapping class group (which as a group of isometries of the upper half-plane can be identified with Out(π1(S))) acts on (X; g) f(x; y) 2 R2j y > 0g equipped with the hyper- by composing the marking with a homeomor- bolic metric ds2 = (dx2 + dy2)=y2. The study of phism of S { the hyperbolic metric on X does P SL(2; Z) and its subgroups via this action has not change. By deforming the metric on X, on occupied legions of mathematicians for well over the other hand, we obtain a neighborhood of the a century. point (X; g) in Teichm¨uller space. We are interested here in the (outer) automor- phism group of a finitely-generated free group. u v Although free groups are the simplest and most fundamental class of infinite groups, their auto- u v u v u u morphism groups are remarkably complex, and v x w v many natural questions about them remain unan- w w swered. We will describe a geometric object On known as Outer space , which was introduced in w w u v [2] to study Out(Fn). u v n The abelianization map Fn ! Z induces a map Out(Fn) ! GL(n; Z), which is an isomor- w phism for n = 2. Thus the upper half plane w used to study P SL(2; Z) can serve equally well u v for Out(F2). For n > 2 the map to GL(n; Z) is x surjective but has a large kernel, so the action w of Out(Fn) on the higher-dimensional homoge- neous space SL(n; R)=SO(n; R) is not proper::: Figure 1: Marked graphs which stabilizers of points are infinite and difficult to are close in Outer space understand. This makes the homogeneous space unsuitable for studying Out(Fn), and something new is needed. To get a space on which Out(Fn) acts, we imi- P SL(2; Z) can also be interpreted as the map- tate the above construction of Teichm¨ullerspace, ping class group of a torus, and the upper half- replacing S by a graph R with one vertex and plane as the Teichm¨ullerspace of the torus. This n edges (a rose with n petals). However, we no shift in viewpoint motivates the construction we longer insist that the marking g be a homeomor- will give here of Outer space. In general, the phism; instead we require only that g be a homo- mapping class group of a closed surface S is the topy equivalence, and we take its target X to be group of isotopy classes of homeomorphisms of S. a finite graph whose edges are isometric to inter- (Recall that two homeomorphisms are isotopic if vals in R. Thus a point in On is a pair (X; g), one can be deformed to the other through a con- where X is a finite metric graph and g : R ! X tinuous family of homeomorphisms.) One way to is a homotopy equivalence. Two pairs (X; g) and describe a point in the Teichm¨ullerspace of S is (X0; g0) are equivalent if X and X0 are isometric as a pair (X; g), where X is a surface equipped and g is homotopic to g0 under some isometry. In 1 2 order to make On finite-dimensional, we also as- where several marked graphs near the red graph sume that the graphs X are connected and have are obtained by folding pairs of edges incident no vertices of valence one or two. Finally, it is to the vertex x together for a small distance. In usually convenient to normalize by assuming the general there are many different possible foldings, sum of the lengths of the edges is equal to one. and this translates to the fact that there is no The mapping class group acts on Teichm¨uller Euclidean coordinate system which describes all space by changing the marking, and the anal- nearby points, i.e. Outer space is not a manifold. ogous statement is true here: an element of Outer space is not too wild, however...it does Out(Fn), represented by a homotopy equivalence have the structure of a locally finite cell com- of R, acts on Outer space by changing only the plex, and it is a theorem that Outer space is con- marking, not the metric graph. A major differ- tractible. It also has the structure of a union ence from Teichm¨ullerspace appears when one of open simplices, each obtained by varying the looks closely at a neighborhood of a point. Te- edge-lengths of a given marked graph (X; g). For ichm¨ullerspace is a manifold. In Outer space n = 2 these simplices can have dimension 1 or 2, points arbitrarily close to a given point (X; g) but not dimension 0, so Outer space is a union of may be of the form (Y; h) with Y not homeo- open triangles identified along open edges (Figure morphic to X. An example is shown in Figure 1, 2). Figure 2: Outer space in rank 2 The stabilizer of a point (X; g) under the ac- these facts are frequently inspired by proofs in the tion of Out(Fn) is isomorphic to the group of analogous settings and use the action of Out(Fn) isometries of the graph X. In particular, it is on Outer space. However, the details are often of a finite group, so the action is proper. Therefore a completely different nature and can vary dra- Outer space serves as an appropriate analog of matically in difficulty, occasionally being easier the homogeneous space used to study a lattice for Out(Fn) but more often easier in at least one in a semisimple Lie group, or of the Teichm¨uller of the other settings. space used to study the mapping class group of a Perhaps the most extensive use of Outer space surface. to date has been for computing algebraic invari- The analogies with lattices and with map- ants of Out(Fn) such as cohomology and Eu- ping class groups have turned out to be quite ler characteristic. Appropriate variations, sub- strong. For example, it has been shown that spaces, quotient spaces and completions of Outer Out(Fn) shares many cohomological properties, space are also used. For example, the fact that basic subgroup structure and many rigidity prop- Out(Fn) acts with finite stabilizers on On implies erties with these classes of groups. The proofs of that a finite-index, torsion-free subgroup Γ acts 3 freely. Therefore the cohomology of Γ is equal the spine intrinsically contains all possible infor- to the cohomology of the quotient On=Γ, and mation about the group! vanishes above the dimension of On. In fact a There are many recent papers on Out(Fn) stronger statement is true: the cohomology must and Outer space, including work on compactifi- also vanish in dimensions below the dimension of cations, metrics and geodesics, dynamics of the On. In order to find the best bound on this van- action, embeddings and fibrations, and connec- ishing (called the virtual cohomological dimension tions to operads and symplectic representation of Out(Fn)), we consider the so-called spine of theory. The interested reader is referred to the Outer space. This is an invariant subspace Kn of three survey articles [1], [3] and [4], for further much lower dimension than On which is a defor- history and for discussion of some of the recent mation retract of the whole space. The dimension developments. of Kn gives a new upper bound for the virtual cohomological dimension, and this upper bound References agrees with a lower bound given by the rank of a [1] Martin R. Bridson and Karen Vogtmann. Automor- free abelian subgroup of Out(Fn). phisms of free groups, surface groups and free abelian Further uses for the spine of Outer space groups. In Benson Farb, editor, Problems on Mapping come from the observation that the quotient Class Groups, number 74 in Proc. Symp. Pure Math., pages 301{316. American Mathematical Society, Prov- Kn=Out(Fn) is compact. As a result one can idence, RI, 2005. show, for example, that the cohomology of [2] Marc Culler and Karen Vogtmann. Moduli of graphs Out(Fn) is finitely-generated in all dimensions and automorphisms of free groups. Invent. Math., and that there are only finitely many conjugacy 84(1):91{119, 1986. classes of finite subgroups. The spine is a simpli- [3] Karen Vogtmann. Automorphisms of free groups and outer space. Geom. Dedicata, 94:1{31, 2002. cial complex on which Out(Fn) acts by permuting [4] Karen Vogtmann. The cohomology of automorphism the simplices, and in fact it was shown by Bridson groups of free groups. In Proceedings of the Interna- and Vogtmann that Out(Fn) is isomorphic to its tional Congress of Mathematicians (Madrid, 2006), full group of simplicial automorphisms. Thus the 2006..
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