For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential.
Hatcher and Vogtmann proved that they are at most exponential, and the
complementary lower bound in the case n=3 was established by Bridson and
Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for
n>4 to the case n=3. In this note we give a shorter, more direct proof of this
last reduction.Comment: Final version, to appear in Annales de l'Institut Fourie

Bridson, Martin R.

Vogtmann, Karen

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The Dehn functions of $Out(F_n)$ and $Aut(F_n)$

Annales de l’institut Fourier (AIF)

ANNALESDEL’INSTITUTFOURIERANNALESDEL’INSTITUT FOURIERMartin R. BRIDSON & Karen VOGTMANNThe Dehn functions of Out(Fn) and Aut(Fn)Tome 62, no 5 (2012), p. 1811-1817.<http://aif.cedram.org/item?id=AIF_2012__62_5_1811_0>© Association des Annales de l’institut Fourier, 2012, tous droitsréservés.L’accès aux articles de la revue « Annales de l’institut Fourier »(http://aif.cedram.org/), implique l’accord avec les conditionsgénérales d’utilisation (http://aif.cedram.org/legal/). Toute re-production en tout ou partie de cet article sous quelque forme quece soit pour tout usage autre que l’utilisation à fin strictement per-sonnelle du copiste est constitutive d’une infraction pénale. Toutecopie ou impression de ce fichier doit contenir la présente mentionde copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Ann. Inst. Fourier, Grenoble62, 5 (2012) 1811-1817THE DEHN FUNCTIONS OF Out(Fn) AND Aut(Fn)by Martin R. BRIDSON & Karen VOGTMANN (*)Abstract. — For n at least 3, the Dehn functions of Out(Fn) and Aut(Fn) areexponential. Hatcher and Vogtmann proved that they are at most exponential, andthe complementary lower bound in the case n = 3 was established by Bridson andVogtmann. Handel and Mosher completed the proof by reducing the lower boundfor n bigger than 3 to the case n = 3. In this note we give a shorter, more directproof of this last reduction.Résumé. — Pour n au moins 3, les fonctions de Dehn de Out(Fn) et Aut(Fn)sont exponentielles. Hatcher et Vogtmann ont montré qu’elles étaient au plus expo-nentielles, et la borne inférieure a été établie par Bridson et Vogtmann dans le casn = 3. Handel et Mosher ont complété la démonstration en ramenant la preuve dela borne inférieure pour n au moins 4 au cas n = 3. Dans cet article, nous donnonsun argument plus direct permettant de passer du cas n = 3 au cas général.Dehn functions provide upper bounds on the complexity of the wordproblem in finitely presented groups. They are examples of filling functions:if a group G acts properly and cocompactly on a simplicial complex X,then the Dehn function of G is asymptotically equivalent to the functionthat provides the optimal upper bound on the area of least-area discs in X,where the bound is expressed as a function of the length of the boundary ofthe disc. This article is concerned with the Dehn functions of automorphismgroups of finitely-generated free groups.Much of the contemporary study of Out(Fn) and Aut(Fn) is based onthe deep analogy between these groups, mapping class groups, and latticesin semisimple Lie groups, particularly SL(n,Z). The Dehn functions ofmapping class groups are quadratic [9], as is the Dehn function of SL(n,Z)if n > 5 (see [10]). In contrast, Epstein et al. [6] proved that the Dehnfunction of SL(3,Z) is exponential. Building on their result, we provedKeywords: Automorphism groups of free groups, Dehn functions.Math. classification: 20F65, 20F28, 53C24, 57S25.(*) Bridson is supported by an EPSRC Senior Fellowship. Vogtmann is supported byNSF grant DMS-0204185.1812 Martin R. BRIDSON & Karen VOGTMANNin [3] that Aut(F3) and Out(F3) also have exponential Dehn functions.Hatcher and Vogtmann [8] established an exponential upper bound on theDehn function of Aut(Fn) and Out(Fn) for all n > 3. The comparison withSL(n,Z) might lead one to suspect that this last result is not optimal forlarge n, but recent work of Handel and Mosher [7] shows that in fact it is:they establish an exponential lower bound by using their general results onquasi-retractions to reduce to the case n = 3.Theorem. — For n > 3, the Dehn functions of Aut(Fn) and Out(Fn)are exponential.This theorem answers Questions 35 and 37 of [4].We learned the contents of [7] from Lee Mosher at Luminy in June 2010and realized that one can also reduce the Theorem to the case n = 3 us-ing a simple observation about natural maps between different-rank Outerspaces and Auter spaces (Lemma 3). The purpose of this note is record thisobservation and the resulting proof of the Theorem.1. DefinitionsLet A be a 1-connected simplicial complex. We consider simplicial loops` : S → A(1), where S is a simplicial subdivision of the circle. A simplicialfilling of ` is a simplicial map L : D → A(2), whereD is a triangulation of the2-disc and L|∂D = `. Such fillings always exist, by simplicial approximation.The filling area of `, denoted AreaA(`), is the least number of triangles inthe domain of any simplicial filling of `. The Dehn function(1) of A isthe least function δA : N → N such that AreaA(`) 6 δA(n) for all loops oflength 6 n in A(1). The Dehn function of a finitely presented group G is theDehn function of any 1-connected 2-complex on which G acts simpliciallywith finite stabilizers and compact quotient. This is well-defined up to thefollowing equivalence relation: functions f, g : N→ N are equivalent if f  gand g  f , where f  g means that there is a constant a > 1 such thatf(n) 6 a g(an + a) + an + a. The Dehn function can be interpreted as ameasure of the complexity of the word problem for G — see [2].Lemma 1. — If A and B are 1-connected simplicial complexes, F : A→B is a simplicial map, and ` is a loop in the 1-skeleton of A, then AreaA(`) >AreaB(F ◦ `).(1)The standard definition of area and Dehn function are phrased in terms of singulardiscs, but this version is ' equivalent.ANNALES DE L’INSTITUT FOURIERDEHN FUNCTION OF Out(Fn) 1813Proof. — If L : D → A is a simplicial filling of `, then F ◦L is a simplicialfilling of F ◦ `, with the same number of triangles in the domain D. Corollary. — Let A,B and C be 1-connected simplicial complexeswith simplicial maps A→ B → C. Let `n be a sequence of simplicial loopsin A whose length is bounded above by a linear function of n, let `n be theimage loops in C and let α(n) = AreaC(`n). Then the Dehn function of Bsatisfies δB(n)  α(n).Proof. — This follows from Lemma 1 together with the observation thata simplicial map does not increase the length of any loop in the 1-skeleton.2. Simplicial complexes associated to Out(Fn) and Aut(Fn)LetKn denote the spine of Outer space, as defined in [5], and Ln the spineof Auter space, as defined in [8]. These are contractible simplicial complexeswith cocompact proper actions by Out(Fn) and Aut(Fn) respectively, sowe may use them to compute the Dehn functions for these groups.Recall from [5] that a marked graph is a finite metric graph Γ togetherwith a homotopy equivalence g : Rn → Γ, where Rn is a fixed graph withone vertex and n loops. A vertex of Kn can be represented either as amarked graph (g,Γ) with all vertices of valence at least three, or as afree minimal action of Fn on a simplicial tree (namely the universal coverof Γ). A vertex of Ln has the same descriptions except that there is achosen basepoint in the marked graph (respected by the marking) or inthe simplicial tree. Note that we allow marked graphs to have separatingedges. Both Kn and Ln are flag complexes, so to define them it sufficesto describe what it means for vertices to be adjacent. In the marked-graphdescription, vertices of Kn (or Ln) are adjacent if one can be obtained fromthe other by a forest collapse (i.e. collapsing each component of a forest toa point).3. Three natural mapsThere is a forgetful map φn : Ln → Kn which simply forgets the base-point; this map is simplicial.Letm < n. We fix an ordered basis for Fn, identify Fm with the subgroupgenerated by the first m elements of the basis, and identify Aut(Fm) withTOME 62 (2012), FASCICULE 51814 Martin R. BRIDSON & Karen VOGTMANNthe subgroup of Aut(Fn) that leaves Fm < Fn invariant and fixes the lastn −m basis elements. We consider two maps associated to this choice ofbasis.First, there is an equivariant augmentation map ι : Lm → Ln whichattaches a bouquet of n−m circles to the basepoint of each marked graphand marks them with the last n − m basis elements of Fn. This map issimplicial, since a forest collapse has no effect on the bouquet of circles atthe basepoint.Secondly, there is a restriction map ρ : Kn → Km which is easiest todescribe using trees. A point in Kn is given by a minimal free simplicialaction of Fn on a tree T with no vertices of valence 2. We define ρ(T ) tobe the minimal invariant subtree for Fm < Fn; more explicitly, ρ(T ) isthe union of the axes in T of all elements of Fm. (Vertices of T that havevalence 2 in ρ(T ) are no longer considered to be vertices.)One can also describe ρ in terms of marked graphs. The chosen embed-ding Fm < Fn corresponds to choosing an m-petal subrose Rm ⊂ Rn. Avertex in Kn is given by a graph Γ marked with a homotopy equivalenceg : Rn → Γ, and the restriction of g to Rm lifts to a homotopy equivalenceĝ : Rm → Γ̂, where Γ̂ is the covering space corresponding to g∗(Fm). Thereis a canonical retraction r of Γ̂ onto its compact core, i.e. the smallest con-nected subgraph containing all nontrivial embedded loops in Γ. Let Γ̂0 bethe graph obtained by erasing all vertices of valence 2 from the compactcore and define ρ(g,Γ) = (r ◦ ĝ, Γ̂0).Lemma 2. — For m < n, the restriction map ρ : Kn → Km is simplicial.Proof. — Any forest collapse in Γ is covered by a forest collapse in Γ̂that preserves the compact core, so ρ preserves adjacency. Lemma 3. — For m < n, the following diagram of simplicial maps com-mutes:Lmι−→ Lnφmy yφnKmρ←− KnProof. — Given a marked graph with basepoint (g,Γ; v) ∈ Ln, themarked graph ι(g,Γ; v) is obtained by attaching n−m loops at v labelledby the elements am+1, . . . , an of our fixed basis for Fn. Then (gn,Γn) :=φn ◦ ι(g,Γ; v) is obtained by forgetting the basepoint, and the cover of(gn,Γn) corresponding to Fm < Fn is obtained from a copy of (g,Γ) (withits labels) by attaching 2(n − m) trees. (These trees are obtained fromthe Cayley graph of Fn as follows: one cuts at an edge labelled aεi , withANNALES DE L’INSTITUT FOURIERDEHN FUNCTION OF Out(Fn) 1815i ∈ {m + 1, . . . , n} and ε = ±1, takes one component of the result, andthen attaches the hanging edge to the basepoint v of Γ.) The effect of ρ isto delete these trees. 4. Proof of the TheoremIn the light of the Corollary and Lemma 3, it suffices to exhibit a se-quence of loops `i in the 1-skeleton of L3 whose lengths are bounded by alinear function of i and whose filling area when projected to K3 grows expo-nentially as a function of i. Such a sequence of loops is essentially describedin [3]. What we actually described there were words in the generators ofAut(F3) rather than loops in L3, but standard quasi-isometric argumentsshow that this is equivalent. More explicitly, the words we considered werewi = T iAT−iBT iA−1T−iB−1 whereT :a1 7→ a21a2a2 7→ a1a2 ,a3 7→ a3A :a1 7→ a1a2 7→ a2 ,a3 7→ a1a3B :a1 7→ a1a2 7→ a2a3 7→ a3a2.To interpret these as loops in the 1-skeleton of L3 (and K3) we note thatA = λ31 andB = ρ32 are elementary transvections and T is the compositionof two elementary transvections: T = λ21 ◦ ρ12. Thus wi is the product of8i+ 4 elementary transvections. There is a (connected) subcomplex of the1-skeleton of L3 spanned by roses (graphs with a single vertex) and Nielsengraphs (which have (n− 2) loops at the base vertex and a further trivalentvertex). We say roses are adjacent if they have distance 2 in this graph.Let I ∈ L3 be the rose marked by the identity map R3 → R3. Each ele-mentary transvection τ moves I to an adjacent rose τI, which is connectedto I by a Nielsen graph Nτ . A composition τ1 . . . τk of elementary transvec-tions gives a path through adjacent roses I, τ1I, τ1τ2I, . . . , τ1τ2 . . . τkI; theNielsen graph connecting σI to στI is σNτ . Thus the word wi correspondsto a loop `i of length 16i + 8 in the 1-skeleton of L3. Theorem A of [3]provides an exponential lower bound on the filling area of φ ◦ `i in K3. The square of maps in Lemma 3 ought to have many uses beyond the onein this note (cf. [7]). We mention just one, for illustrative purposes. This isa special case of the fact that every infinite cyclic subgroup of Out(Fn) isquasi-isometrically embedded [1].Proposition. — The cyclic subgroup of Out(Fn) generated by anyNielsen transformation (elementary transvection) is quasi-isometrically em-bedded.TOME 62 (2012), FASCICULE 51816 Martin R. BRIDSON & Karen VOGTMANNProof. — Each Nielsen transformation is in the image of the mapΦ: Aut(F2)→ Aut(Fn)→ Out(Fn)given by the inclusion of a free factor F2 < Fn. Thus it suffices to provethat if a cyclic subgroup C = 〈c〉 < Aut(F2) has infinite image in Out(F2),then t 7→ Φ(ct) is a quasi-geodesic. This is equivalent to the assertion thatsome (hence any) C-orbit in Kn is quasi-isometrically embedded, where Cacts on Kn as Φ(C) and Kn is given the piecewise Euclidean metric whereall edges have length 1.K2 is a tree and C acts on K2 as a hyperbolic isometry, so the C-orbitsin K2 are quasi-isometrically embedded. For each x ∈ L2, the C-orbit ofφ2(x) is the image of the quasi-geodesic t 7→ ct.φ2(x) = φ2(ct.x). We factorφ2 as a composition of C-equivariant simplicial maps L2ι→ Knφn→ K2, as inLemma 3, to deduce that the C-orbit of φnι(x) in Kn is quasi-isometricallyembedded. A slight variation on the above argument shows that if one lifts a freegroup of finite index Λ < Out(F2) to Aut(F2) and then maps it to Out(Fn)by choosing a free factor F2 < Fn, then the inclusion Λ ↪→ Out(Fn) will bea quasi-isometric embedding.BIBLIOGRAPHY[1] E. Alibegovic, “Translation lengths in Out(Fn)”,Geom. Dedicata 92 (2002), p. 87-93.[2] M. R. Bridson, “The geometry of the word problem”, in Invitations to geometryand topology, Oxf. Grad. Texts Math., vol. 7, Oxford Univ. Press, Oxford, 2002,p. 29-91.[3] M. R. Bridson & K. Vogtmann, “On the geometry of the automorphism groupof a free group”, Bull. Math Londres. Soc. 27 (1995), p. 544-552.[4] ———, “Automorphism groups of free groups, surface groups and free abeliangroups”, in Problems on mapping class groups and related topics, Proc. Sympos.Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, p. 301-316.[5] M. Culler & K. Vogtmann, “Moduli of graphs and automorphisms of free groups”,Invent. Math. 84 (1986), no. 1, p. 91-119.[6] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson& W. P. Thurston, Word processing in groups, Jones and Bartlett Publishers,Boston, MA, 1992, xii+330 pages.[7] M. Handel & L. Mosher, “Lipschitz retraction and distortion for subgroups ofOut(Fn)”, arXiv:1009.5018, 2010.[8] A. Hatcher & K. Vogtmann, “Isoperimetric inequalities for automorphism groupsof free groups”, Pacific J. Math. 173 (1996), no. 2, p. 425-441.ANNALES DE L’INSTITUT FOURIERDEHN FUNCTION OF Out(Fn) 1817[9] L. Mosher, “Mapping class groups are automatic”, Ann. of Math. (2) 142 (1995),no. 2, p. 303-384.[10] R. Young, “The Dehn function of SL(n;Z)”, arXiv:0912.2697v1, 2009.Manuscrit reçu le 7 décembre 2010,accepté le 8 février 2011.Martin R. BRIDSONMathematical Institute24-29 St Giles’Oxford OX1 3LB (U.K.)bridson@maths.ox.ac.ukKaren VOGTMANNCornell UniversityDepartment of MathematicsIthaca NY 14853 (USA)vogtmann@math.cornell.eduTOME 62 (2012), FASCICULE 5

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The Dehn functions of Out(F_n) and Aut(F_n)

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