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Martin R. BRIDSON & Karen VOGTMANN
The Dehn functions of Out(Fn) and Aut(Fn)
Tome 62, no 5 (2012), p. 1811-1817.
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Ann. Inst. Fourier, Grenoble
62, 5 (2012) 1811-1817
THE 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) 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 bigger than 3 to the case n = 3. In this note we give a shorter, more direct
proof 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 cas
n = 3. Handel et Mosher ont complété la démonstration en ramenant la preuve de
la borne inférieure pour n au moins 4 au cas n = 3. Dans cet article, nous donnons
un argument plus direct permettant de passer du cas n = 3 au cas général.
Dehn functions provide upper bounds on the complexity of the word
problem 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 function
that 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 of
the disc. This article is concerned with the Dehn functions of automorphism
groups of finitely-generated free groups.
Much of the contemporary study of Out(Fn) and Aut(Fn) is based on
the deep analogy between these groups, mapping class groups, and lattices
in semisimple Lie groups, particularly SL(n,Z). The Dehn functions of
mapping 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 Dehn
function of SL(3,Z) is exponential. Building on their result, we proved
Keywords: Automorphism groups of free groups, Dehn functions.
Math. classification: 20F65, 20F28, 53C24, 57S25.
(*) Bridson is supported by an EPSRC Senior Fellowship. Vogtmann is supported by
NSF grant DMS-0204185.
1812 Martin R. BRIDSON & Karen VOGTMANN
in [3] that Aut(F3) and Out(F3) also have exponential Dehn functions.
Hatcher and Vogtmann [8] established an exponential upper bound on the
Dehn function of Aut(Fn) and Out(Fn) for all n > 3. The comparison with
SL(n,Z) might lead one to suspect that this last result is not optimal for
large 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 on
quasi-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 2010
and realized that one can also reduce the Theorem to the case n = 3 us-
ing a simple observation about natural maps between different-rank Outer
spaces and Auter spaces (Lemma 3). The purpose of this note is record this
observation and the resulting proof of the Theorem.
1. Definitions
Let A be a 1-connected simplicial complex. We consider simplicial loops
` : S → A(1), where S is a simplicial subdivision of the circle. A simplicial
filling of ` is a simplicial map L : D → A(2), whereD is a triangulation of the
2-disc and L|∂D = `. Such fillings always exist, by simplicial approximation.
The filling area of `, denoted AreaA(`), is the least number of triangles in
the domain of any simplicial filling of `. The Dehn function(1) of A is
the least function δA : N → N such that AreaA(`) 6 δA(n) for all loops of
length 6 n in A(1). The Dehn function of a finitely presented group G is the
Dehn function of any 1-connected 2-complex on which G acts simplicially
with finite stabilizers and compact quotient. This is well-defined up to the
following equivalence relation: functions f, g : N→ N are equivalent if f  g
and g  f , where f  g means that there is a constant a > 1 such that
f(n) 6 a g(an + a) + an + a. The Dehn function can be interpreted as a
measure 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 singular
discs, but this version is ' equivalent.
ANNALES DE L’INSTITUT FOURIER
DEHN FUNCTION OF Out(Fn) 1813
Proof. — If L : D → A is a simplicial filling of `, then F ◦L is a simplicial
filling of F ◦ `, with the same number of triangles in the domain D. 
Corollary. — Let A,B and C be 1-connected simplicial complexes
with simplicial maps A→ B → C. Let `n be a sequence of simplicial loops
in A whose length is bounded above by a linear function of n, let `n be the
image loops in C and let α(n) = AreaC(`n). Then the Dehn function of B
satisfies δB(n)  α(n).
Proof. — This follows from Lemma 1 together with the observation that
a 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 spine
of Auter space, as defined in [8]. These are contractible simplicial complexes
with cocompact proper actions by Out(Fn) and Aut(Fn) respectively, so
we may use them to compute the Dehn functions for these groups.
Recall from [5] that a marked graph is a finite metric graph Γ together
with a homotopy equivalence g : Rn → Γ, where Rn is a fixed graph with
one vertex and n loops. A vertex of Kn can be represented either as a
marked graph (g,Γ) with all vertices of valence at least three, or as a
free minimal action of Fn on a simplicial tree (namely the universal cover
of Γ). A vertex of Ln has the same descriptions except that there is a
chosen basepoint in the marked graph (respected by the marking) or in
the simplicial tree. Note that we allow marked graphs to have separating
edges. Both Kn and Ln are flag complexes, so to define them it suffices
to describe what it means for vertices to be adjacent. In the marked-graph
description, vertices of Kn (or Ln) are adjacent if one can be obtained from
the other by a forest collapse (i.e. collapsing each component of a forest to
a point).
3. Three natural maps
There 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 subgroup
generated by the first m elements of the basis, and identify Aut(Fm) with
TOME 62 (2012), FASCICULE 5
1814 Martin R. BRIDSON & Karen VOGTMANN
the subgroup of Aut(Fn) that leaves Fm < Fn invariant and fixes the last
n −m basis elements. We consider two maps associated to this choice of
basis.
First, there is an equivariant augmentation map ι : Lm → Ln which
attaches a bouquet of n−m circles to the basepoint of each marked graph
and marks them with the last n − m basis elements of Fn. This map is
simplicial, since a forest collapse has no effect on the bouquet of circles at
the basepoint.
Secondly, there is a restriction map ρ : Kn → Km which is easiest to
describe using trees. A point in Kn is given by a minimal free simplicial
action of Fn on a tree T with no vertices of valence 2. We define ρ(T ) to
be the minimal invariant subtree for Fm < Fn; more explicitly, ρ(T ) is
the union of the axes in T of all elements of Fm. (Vertices of T that have
valence 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. A
vertex in Kn is given by a graph Γ marked with a homotopy equivalence
g : Rn → Γ, and the restriction of g to Rm lifts to a homotopy equivalence
ĝ : Rm → Γ̂, where Γ̂ is the covering space corresponding to g∗(Fm). There
is 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 be
the graph obtained by erasing all vertices of valence 2 from the compact
core 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 Γ̂
that preserves the compact core, so ρ preserves adjacency. 
Lemma 3. — For m < n, the following diagram of simplicial maps com-
mutes:
Lm
ι−→ Ln
φm
y yφn
Km
ρ←− Kn
Proof. — Given a marked graph with basepoint (g,Γ; v) ∈ Ln, the
marked graph ι(g,Γ; v) is obtained by attaching n−m loops at v labelled
by 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) (with
its labels) by attaching 2(n − m) trees. (These trees are obtained from
the Cayley graph of Fn as follows: one cuts at an edge labelled aεi , with
ANNALES DE L’INSTITUT FOURIER
DEHN FUNCTION OF Out(Fn) 1815
i ∈ {m + 1, . . . , n} and ε = ±1, takes one component of the result, and
then attaches the hanging edge to the basepoint v of Γ.) The effect of ρ is
to delete these trees. 
4. Proof of the Theorem
In 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 a
linear 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 described
in [3]. What we actually described there were words in the generators of
Aut(F3) rather than loops in L3, but standard quasi-isometric arguments
show that this is equivalent. More explicitly, the words we considered were
wi = T iAT−iBT iA−1T−iB−1 where
T :

a1 7→ a21a2
a2 7→ a1a2 ,
a3 7→ a3
A :

a1 7→ a1
a2 7→ a2 ,
a3 7→ a1a3
B :

a1 7→ a1
a2 7→ a2
a3 7→ a3a2
.
To interpret these as loops in the 1-skeleton of L3 (and K3) we note that
A = λ31 andB = ρ32 are elementary transvections and T is the composition
of two elementary transvections: T = λ21 ◦ ρ12. Thus wi is the product of
8i+ 4 elementary transvections. There is a (connected) subcomplex of the
1-skeleton of L3 spanned by roses (graphs with a single vertex) and Nielsen
graphs (which have (n− 2) loops at the base vertex and a further trivalent
vertex). 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 connected
to 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; the
Nielsen graph connecting σI to στI is σNτ . Thus the word wi corresponds
to 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 one
in this note (cf. [7]). We mention just one, for illustrative purposes. This is
a special case of the fact that every infinite cyclic subgroup of Out(Fn) is
quasi-isometrically embedded [1].
Proposition. — The cyclic subgroup of Out(Fn) generated by any
Nielsen transformation (elementary transvection) is quasi-isometrically em-
bedded.
TOME 62 (2012), FASCICULE 5
1816 Martin R. BRIDSON & Karen VOGTMANN
Proof. — Each Nielsen transformation is in the image of the map
Φ: Aut(F2)→ Aut(Fn)→ Out(Fn)
given by the inclusion of a free factor F2 < Fn. Thus it suffices to prove
that 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 that
some (hence any) C-orbit in Kn is quasi-isometrically embedded, where C
acts on Kn as Φ(C) and Kn is given the piecewise Euclidean metric where
all edges have length 1.
K2 is a tree and C acts on K2 as a hyperbolic isometry, so the C-orbits
in K2 are quasi-isometrically embedded. For each x ∈ L2, the C-orbit of
φ2(x) is the image of the quasi-geodesic t 7→ ct.φ2(x) = φ2(ct.x). We factor
φ2 as a composition of C-equivariant simplicial maps L2
ι→ Kn
φn→ K2, as in
Lemma 3, to deduce that the C-orbit of φnι(x) in Kn is quasi-isometrically
embedded. 
A slight variation on the above argument shows that if one lifts a free
group 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 be
a 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 geometry
and 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 group
of a free group”, Bull. Math Londres. Soc. 27 (1995), p. 544-552.
[4] ———, “Automorphism groups of free groups, surface groups and free abelian
groups”, 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 of
Out(Fn)”, arXiv:1009.5018, 2010.
[8] A. Hatcher & K. Vogtmann, “Isoperimetric inequalities for automorphism groups
of free groups”, Pacific J. Math. 173 (1996), no. 2, p. 425-441.
ANNALES DE L’INSTITUT FOURIER
DEHN 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. BRIDSON
Mathematical Institute
24-29 St Giles’
Oxford OX1 3LB (U.K.)
bridson@maths.ox.ac.uk
Karen VOGTMANN
Cornell University
Department of Mathematics
Ithaca NY 14853 (USA)
vogtmann@math.cornell.edu
TOME 62 (2012), FASCICULE 5


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

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