A DENDROLOGICAL PROOF OF THE SCOTT CONJECTURE

FOR AUTOMORPHISMS OF FREE GROUPS

Damien Gaboriau, Gilbert Levitt, Martin Lustig

Introduction.

In this note we give a short alternative proof of the following much celebrated result of Bestvina-Handel [BH], conjectured earlier by Scott:

Theorem 1. For every automorphism α of a free group F_{n} of rank n ≥ 1 the fixed subgroup Fix α = {w ∈ F_{n} | α(w) = w} has rank rk Fix α ≤ n.

After a preliminary version of this note was written, other alternative proofs of this result have been given in [Se] and in [Pa]. A stronger inequality, which also takes into account infinite fixed words, is given in [GJLL] (see the remark at the end of this paper).

The essential ingredients in our proof are the following two theorems. The first one, proved by the third author in [Lu], is a fixed point theorem for the action of outer automorphisms of F_{n} on the compactification of Culler-Vogtmann’s outer space. It may be stated as follows.

Theorem 2 [Lu]. Given any automorphism α of F_{n}, there exists a nontrivial minimal action of F_{n} on an R-tree T, with trivial edge stabilizers, whose length function ℓ: F_{n} → R^{+} satisfies ℓ ∘ α = λℓ for some λ ≥ 1. If λ = 1, then T may be taken to be a simplicial tree.

Since the paper [Lu] was written, short proofs for the existence of a fixed point have been obtained. As none of them is presently published, we give one in an Appendix, establishing triviality of edge stabilizers (a key point in our approach).

The second theorem we use here is due to the first two authors [GL]. It is an inequality about branch points in an R-tree equipped with a small action of F_{n}. We state the special case that will be needed.

Theorem 3 [GL]. Let T be an R-tree with a nontrivial, minimal F_{n}-action whose edge stabilizers are all trivial. Given p_{1}, …, p_{m} ∈ T belonging to distinct orbits, the isotropy subgroups Stab p_{i} satisfy the inequality

∑_{i=1}^{m}(rk Stab p_{i} − \frac{1}{2}) ≤ n − 1.

In particular, rk Stab p ≤ n − 1 for every p ∈ T.

Typeset by AMS-TEX

This follows directly from Theorem III.2 of [GL]: since edge stabilizers are trivial, we have v_{1}(p) > 0 for every p ∈ T.

Our proof of the Scott conjecture consists of two parts. In Part 1, we try to prove Theorem 1 by induction on n, using Theorem 2 and analyzing the action on T thanks to Theorem 3. We succeed in all cases but one. In the last case we obtain

rk Fix α = rk Fix α^{a} + rk Fix (i_{u} ∘ α^{a}),

where α^{a} is some automorphism of F_{n−1} and i_{u} is conjugation by u ∈ F_{n−1}: (i_{u} ∘ α^{a})(g) = uα^{a}(g)u^{−1}.

This makes it necessary to study several automorphisms simultaneously. Say that two automorphisms α, β of F_{n} are similar if there exists c ∈ F_{n} such that β = i_{c} ∘ α ∘ (i_{c})^{−1}. In other words, one has β(g) = c α(c^{−1}gc)c^{−1} for every g ∈ F_{n}. Notice that similar automorphisms induce the same outer automorphism and have conjugate fixed subgroups.

In Part 2 we extend the analysis performed in Part 1, so as to prove by induction on n the following strengthening of Theorem 1:

Theorem 1’. Let α_{0}, …, α_{k} be automorphisms of F_{n} representing the same outer automorphism φ and belonging to distinct similarity classes. Then

∑_{i=0}^{k}(rk Fix α_{i} − 1) ≤ n − 1.

(This theorem is only superficially stronger than Theorem 1. It follows by applying Theorem 1 to the automorphism of F_{n} ∗ F_{k} equal to α_{0} on F_{n} and sending the i-th generator t_{i} of F_{k} to t_{i}u_{i} where α_{i} = i_{u_{i}} ∘ α_{0}.)

In Part 3 we discuss equality in Theorems 1 and 1’.

1. Analyzing a single automorphism.

We try to prove Theorem 1 by induction on n, noting that it is trivial for n = 1. Let α be an automorphism of F_{n}, and let T be given by Theorem 2. If λ = 1, then T is simplicial. In this case we require that F_{n} act on T without inversions, and we choose T so as to minimize the number of edges of the quotient graph Γ = T/F_{n}.

It follows directly (as in [Lu, §4]) from the uniqueness result for non-abelian actions (see [CM]) that there exists a homothety H: T → T with stretching factor λ, which commutes with α in the sense that for every w ∈ F_{n} one has

α(w)H = Hw  (1)

as maps from T to T.

1.1 If H has no fixed point, then rk Fix α ≤ 1.

Let w ∈ Fix α, w ≠ 1. Consider its characteristic set C_{w}: it is the axis of w if w acts as a hyperbolic isometry, or the unique fixed point of w if w is elliptic (w has only one fixed point because edge stabilizers are trivial). Since H commutes with w, it preserves C_{w}. This is possible only if λ = 1 and H, w are hyperbolic isometries with the same axis. This implies rk Fix α ≤ 1 since otherwise some nontrivial commutator would have a line of fixed points.

1.2 If H has a fixed point Q, then Stab Q is α-invariant.

¿From wQ = Q it follows α(w)Q = α(w)HQ = HwQ = Q.

We write α^{Q} for the automorphism of Stab Q induced by α (note that Stab Q is also α^{−1}-invariant).

1.3 If Q is the only fixed point of H, then Stab Q contains Fix α.

¿From α(w) = w it follows HwQ = α(w)HQ = wQ, so that wQ = Q if H has only one fixed point.

1.4 If H has at most one fixed point, for instance if λ ≠ 1, we get rk Fix α ≤ n − 1 by applying the induction hypothesis to α^{Q} (recall that Stab Q has rank ≤ n − 1 by Theorem 3).

1.5 From now on we assume that H is an isometry which has more than one fixed point. Recall that T is a simplicial tree, chosen to minimize the number of edges of Γ = T/F_{n}. Let e = [a, b] be an edge fixed (pointwise) by H.

1.6 Under the assumptions of 1.5, the graph Γ has only one edge.

Let T′ be the tree obtained by collapsing each component of the orbit of e to a point. This orbit is preserved by H, so that H induces an isometry H′ of T′ satisfying the commutation equation (1) with the induced F_{n}-action on T′. If Γ has more than one edge, the action of F_{n} on T′ is nontrivial. This contradicts the choice of T.

1.7 First assume that Γ is a segment. Then Bass-Serre theory gives a nontrivial decomposition F_{n} = Stab a ∗ Stab b. This decomposition is α-invariant by 1.2, so that Fix α = Fix α^{a} ∗ Fix α^{b}. The result follows again by induction.

1.8 Finally, we assume that Γ is a loop. Then F_{n} = (Stab a)∗ < t >, where t is any element such that t(a) = b. Note that α(t)a = α(t)Ha = Hta = b, so that α(t) = tu with u ∈ Stab a. If t may be chosen with α(t) = t, we have an α-invariant decomposition as before.

1.9 Otherwise α(t) = tu for some u ∈ Stab a that cannot be written u = vα(v^{−1}) with v ∈ Stab a. Direct computation, or Bass-Serre theory applied to the action of Fix α on Fix H, yields Fix α = Fix α^{a} ∗ t Fix (i_{u} ∘ α^{a}) t^{−1}, so that the induction breaks down here. This method only shows rk Fix α ≤ 2^{n−1}.

2. Analyzing several automorphisms.

Let φ be an outer automorphism of F_{n}. Let T and λ be as in the beginning of Part 1, so that ℓ ∘ φ = λℓ. If λ = 1, recall that T is chosen so as to minimize the number of edges of Γ.

For any α ∈ Aut (F_{n}) representing φ, there is a homothety H_{α}: T → T such that α(w)H_{α} = H_{α}w (w ∈ F_{n}).

2.1 If β = i_{h} ∘α is another representative of φ, we may (and will) take H_{β} = hH_{α}. Since F_{n} acts on T with trivial edge stabilizers, no nondegenerate arc e ⊂ T may be fixed pointwise by both H_{α} and H_{β} if h ≠ 1.

2.2 If β = i_{c} ∘ α ∘ (i_{c})^{−1} is similar to α, we get H_{β} = cα(c^{−1})H_{α} = cH_{α}c^{−1}. In particular Fix H_{β} = cFix H_{α}.

2.3 Let α_{0}, …, α_{k} be representatives of φ belonging to distinct similarity classes. Denoting r(α) = rk Fix α − 1 we now prove the following inequality by induction on n:

∑_{i=0}^{k} r(α_{i}) ≤ n − 1.

It is clear if n = 1, so we assume n ≥ 2. By 1.1 we may assume that each H_{i} = H_{α_{i}} has at least one fixed point.

2.4 Suppose Q ∈ T is fixed by both H_{i} and H_{j} (i ≠ j). Let α_{i}^{Q} and α_{j}^{Q} be the automorphisms of Stab Q induced by α_{i} and α_{j} (see 1.2).

If rk Stab Q ≥ 2, then α_{i}^{Q}, α_{j}^{Q} represent the same outer automorphism of Stab Q and belong to distinct similarity classes in Aut (Stab Q).

If α_{j} = i_{h} ∘ α_{i}, we have h ∈ Stab Q because H_{j} = hH_{i} (see 2.1). Now suppose there exists v ∈ Stab Q such that α_{j}(g) = vα_{i}(v^{−1}gv)v^{−1} for all g ∈ Stab Q. Then

hα_{i}(g)h^{−1} = α_{j}(g) = vα_{i}(v^{−1})α_{i}(g)α_{i}(v)v^{−1}

for g ∈ Stab Q. Since Stab Q has rank ≥ 2, we deduce h = vα_{i}(v^{−1}) so that α_{j}(g) = vα_{i}(v^{−1}gv)v^{−1} holds for every g ∈ F_{n}. This is a contradiction since α_{i} and α_{j} are not similar.

2.5 First assume that each H_{i} has exactly one fixed point Q_{i} (e.g. if λ ≠ 1). Replacing each α_{i} be a similar automorphism, 2.2 lets us assume that for i ≠ j either Q_{i} = Q_{j} or Q_{i}, Q_{j} belong to different orbits. Let Q ⊂ T be the set of all points Q_{i}, and π: {0, …, k} → Q the map taking i to Q_{i}.

We then write

∑_{i=0}^{k} r(α_{i}) = ∑_{i=0}^{k} r(α_{i}^{Q_{i}}) = ∑_{Q∈Q} ∑_{i∈π^{−1}(Q)} r(α_{i}^{Q_{i}}) ≤ ∑_{Q∈Q}(rk Stab Q − 1) < n − 1.

The first equality comes from 1.3. The first inequality is clear for points Q with rk Stab Q ≤ 1. For other points it follows from the induction hypothesis thanks to 2.4. The second inequality is Theorem 3 since different points of Q are in different F_{n}-orbits.

2.6 If some H_{i} (say H_{0}) fixes (pointwise) an edge e = [a, b], then Γ = T/F_{n} has only one edge by 1.6. Using 2.1 and 2.2 we see that for i > 0 the map H_{i} has only one fixed point Q_{i}. By 2.2 we may assume Q_{i} = a or b (unless Q_{i} is the midpoint of an edge, but then Fix α_{i} is trivial by 1.3).

2.7 First assume that Γ is a segment. Using 1.7 we write r(α_{0}) = 1+r(α_{0}^{a})+r(α_{0}^{b}). Taking π to be the obvious map from {1, …, k} to {a, b}, we then get as in 2.5:

∑_{i=0}^{k} r(α_{i}) = 1 + r(α_{0}^{a}) + ∑_{i∈π^{−1}(a)} r(α_{i}^{a}) + r(α_{0}^{b}) + ∑_{i∈π^{−1}(b)} r(α_{i}^{b})

≤ 1 + rk Stab a − 1 + rk Stab b − 1 = n − 1.

2.8 If Γ is a loop, we assume Q_{i} = a for every i ≥ 1. We study α_{0} as in 1.8 and 1.9. In the situation of 1.8 we argue as in 2.7. In the situation of 1.9 we get:

∑_{i=0}^{k} r(α_{i}) = ∑_{i=0}^{k} r(α_{i}^{a}) + r(i_{u} ∘ α_{0}^{a}) + 1.

There is nothing to prove if Stab a has rank 1. Otherwise we argue as follows. The automorphisms α_{0}^{a}, α_{1}^{a}, …, α_{k}^{a}, i_{u} ∘α_{0}^{a} represent the same outer automorphism of Stab a. The inductive proof will then be complete if we show that no two of them are similar.

By 2.4 we only need to check that i_{u} ∘ α_{0}^{a} is not similar to any of the others. Arguing as in 2.4 we see that it is not similar to α_{0}^{a} since otherwise we would be in the situation of 1.8. Now we note that i_{u} ∘ α_{0} is similar to α_{0} in Aut (F_{n}) since i_{u} ∘ α_{0} = (i_{t})^{−1} ∘ α_{0} ∘ i_{t}. It follows that i_{u} ∘ α_{0} and α_{i} are not similar for i ≥ 1. Using 2.4 we see that i_{u} ∘ α_{0}^{a} and α_{i}^{a} are not similar.

3. Equality.

Suppose equality holds in Theorem 1’. Perform the analysis of Part 2. We have shown that case 2.5 is impossible. In case 2.7 (resp. 2.8), equality holds in Stab a and Stab b (resp. in Stab a). Furthermore both α_{0}^{a} and α_{0}^{b} (resp. α_{0}^{a} and i_{u} ∘ α_{0}^{a}) fix nontrivial elements. An easy induction then yields the following result:

Theorem. Let α_{0}, …, α_{k} be automorphisms of F_{n} representing the same outer automorphism φ and belonging to distinct similarity classes. If ∑_{i=0}^{k}(rk Fix α_{i} − 1) = n − 1 and n ≥ 2, then some α_{i} with rk Fix α_{i} ≥ 2 fixes an element of a free basis of F_{n}.

Corollary [CT]. If α ∈ Aut (F_{n}) satisfies rk Fix F_{n} = n, then α fixes an element of a free basis of F_{n}.

If n ≥ 2, one may require in this corollary that the fixed basis element belong to an α-invariant free factor of rank n − 1.

Remark. In connection with the paper [CL], the third author has conjectured some years ago the following strengthening of Theorem 1’:

∑_{i=0}^{k}(rk Fix α_{i} − 1 + \frac{1}{2}a(α_{i})) ≤ n − 1

where a(α_{i}) denotes the number of attractive fixed points at ∞ of α_{i} (see [CL]). A proof of this conjecture has been obtained in joint work with A. Jaeger, see [GJLL]. The paper [GJLL] uses the analysis presented here, but it is much more involved and its main emphasis is on new geometric techniques.

Appendix.

In this appendix we provide a proof of Theorem 2. We follow closely the argument sketched in [BF] for irreducible automorphisms, using the existence of (not necessarily stable) train track representatives as given by Theorem 5.12 of [BH], but not the main body of Bestvina-Handel’s approach (indivisible Nielsen paths, etc.).

Let ϕ be the outer automorphism associated to α. By [BH, Theorem 5.12], we may represent ϕ by a relative train track map f : τ → τ, where τ is a finite graph without vertices of valence 1 with π_{1}τ ≃ F_{n}, and f respects a maximal filtration τ_{0} ⊂ τ_{1} ⊂ … ⊂ τ_{r} = τ. Furthermore there exists a nonzero length assignment L(e) ≥ 0 for the edges e of τ, and a constant λ ≥ 1, such that for p ≥ 1 one has L(f^{p}(e)) = λ^{p}L(e) for every edge e and L(f(γ)) ≤ L(γ) for every loop γ; here L( .) denotes the length of the shortest representative in the homotopy class (relative to endpoints). The function L comes from the Perron-Frobenius eigenvector of the (irreducible) transition matrix M(f|_{τ\τ_{r−1}}) and λ is the corresponding eigenvalue, see [BH].

If λ = 1, then the transition matrix is a permutation matrix, and we let T be the simplicial R-tree T_{0} obtained by lifting L to the universal covering of τ.

If λ > 1, we consider the length function ℓ_{0} of the action of F_{n} on T_{0}, the nonincreasing sequence of length functions ℓ_{n} = \frac{1}{λ^{n}} (ℓ ∘ α^{n}), and the limit ℓ_{∞}. This limit satisfies ℓ_{∞} ∘ α = ℓ_{∞}. By [CM] either it is identically 0, or it is the length function of an action of F_{n} on an R-tree T whose edge stabilizers are cyclic or trivial. There remains to show that ℓ_{∞} is not identically 0, and that edge stabilizers are trivial.

Let e be an edge with L(e) > 0. Since L(f^{p}(e)) = λ^{p}L(e) tends to infinity and τ is finite, we can find a reduced loop γ which is a subpath of some f^{p}(e), with L(γ) arbitrarily large. The image f(γ) is a loop, which is reduced except possibly at its basepoint: f(γ) = δγ′δ^{−1}, where γ′ is a reduced loop. Since f is a homotopy equivalence of a finite graph, there is a bound C, depending only on f, for the simplicial length of δ. Setting C′ = C max(L(e_{i})) we obtain ℓ(α(γ)) = L(γ′) ≥ λL(γ)−2C′ = λℓ(γ)−2C′. Similarly we get ℓ(α(α^{q}(γ)) ≥ λℓ(α^{q}(γ))−2C′ for q ≥ 1, hence ℓ_{∞}(γ) > 0 if L(γ) is large enough. This proves that ℓ_{∞} is not identically 0.

To prove triviality of edge stabilizers, we may assume that the action of F_{n} on T is minimal. Let H be the homothety introduced in Part 1. Let c ⊂ T be a segment fixed by some nontrivial w ∈ F_{n}. Let s ≥ 1 be the largest integer such that w is an s-th power. Since the length of H^{k}(c) grows arbitrary large with k and there exists a finite collection of segments whose union meets every orbit, we can find, for sufficiently large k, disjoint non-degenerate subarcs c_{0}, …, c_{s} of H^{k}(c) such that c_{i} = v_{i}c_{0} for some v_{i} ∈ F_{n} (i = 1, …, s).

The element w′ = α^{k}(w) fixes H^{k}(c). Since w′, v_{1}, …, v_{s} all have different actions on c_{0}, there exists i such that v_{i} and w′ do not generate a cyclic subgroup of F_{n}. Then w′ and v_{i}^{−1}w′v_{i} generate a free subgroup of rank 2 which fixes c_{i} pointwise, contrary to the fact that edge stabilizers are cyclic or trivial.

References

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[GJLL] D. Gaboriau, A. Jaeger, G. Levitt, M. Lustig, An index for counting fixed points of automorphisms of free groups, preprint.

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[Lu] M. Lustig, Automorphisms, train tracks and non-simplicial R-tree actions, preprint.

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D.G.: Umpa, cnrs umr 128, Ens Lyon, 69364 Lyon Cedex 07, France. e-mail:gaboriau@umpa.ens-lyon.fr

G.L.: Laboratoire Émile Picard, cnrs umr 5580, Université Paul Sabatier, 31062 Toulouse Cedex, France. e-mail: levitt@picard.ups-tlse.fr

M.L.: Institut für Mathematik, Ruhr-Universität Bochum, Postfach 10 21 48, 4630 Bochum 1, Germany. e-mail: Martin.Lustig@rz.ruhr-uni-bochum.de