THE RANK OF ACTIONS ON R-TREES

Damien GABORIAU and Gilbert LEVITT

Abstract.

For n ≥ 2, let F_{n} denote the free group of rank n. We define a total branching index i for a minimal small action of F_{n} on an R-tree. We show i ≤ 2n − 2, with equality if and only if the action is geometric. We thus recover Jiang’s bound 2n − 2 for the number of orbits of branch points of free F_{n}-actions, and we extend it to very small actions (i.e. actions which are limits of free actions).

The Q-rank of a minimal very small action of F_{n} is bounded by 3n − 3, equality being possible only if the action is free simplicial. There exists a free action of F_{3} such that the values of the length function do not lie in any finitely generated

subgroup of R.

The boundary of Culler-Vogtmann’s outer space Y_{n} has topological dimension 3n − 5.

INTRODUCTION AND STATEMENT OF RESULTS.

Various problems from geometry and group theory lead to isometric group actions on R-trees. An R-tree is a path-connected metric space in which every arc is isometric to an interval of R. See the surveys [Sh 1], [Sh 2], [Mo] and the papers [AB], [CM] for basic results about R-trees.

These actions on R-trees are most often small: no edge stabilizer contains a free non-abelian subgroup. Following work of Rips, it is now known that hyperbolic groups admitting nontrivial small actions on R-trees have nontrivial splittings (see [BF 2] for precise statements and corollaries).

Small actions of a given finitely generated group G determine a closed subspace in the space of all length functions on G. This subspace is often infinite dimensional [CL, Theorem 9.8]. Bestvina-Feighn have proved a finiteness theorem for reduced simplicial small actions [BF 1].

In this paper we consider actions of F_{n}, the free group of rank n. We obtain finiteness results about branch points, rank, and Culler-Vogtmann’s outer space. Our results apply to small actions, and to very small actions.

Recall (Cohen-Lustig [CL]) that a small action of F_{n} on an R-tree is very small if for every nontrivial g ∈ F_{n} the fixed subtree Fix(g) is equal to Fix(g^{p}) for p ≥ 2 (no obtrusive powers) and Fix(g) is isometric to a subset of R (no fixed triods).

Outer space Y_{n} consists of (projective classes of length functions of) free simplicial actions of F_{n}, and its closure consists precisely of very small actions [BF 3]. In particular, an action is very small if and only if it is a limit of free actions.

Typeset by AMS-TEX

Let T be a small F_{n}-tree (i.e. an R-tree equipped with a small action of F_{n}). We always assume that T is minimal (there is no proper invariant subtree).

Let x ∈ T be a branch point (i.e. a point such that T \ {x} has at least 3 components). In Part III we define an index i(x) in terms of the isotropy subgroup Stab(x) and its action on the set of directions π_{0}(T \ {x}), by

i(x) = 2 rk Stab(x) + v_{1}(x) − 2,

where v_{1}(x) is the number of Stab(x)-orbits of directions with trivial stabilizer; it turns out that i(x) ∈ N.

The index i(x) depends only on the orbit O = F_{n}(x) and we define the index of T as

i(T) = ∑_{O∈T/F_{n}} i(O).

Theorem III.2. Let T be a small minimal F_{n}-tree. Then i(T) ≤ 2n − 2.

If the action is very small, the index of every branch point is positive. We then get:

Corollary III.3. Let T be a very small minimal F_{n}-tree. The number b of orbits of branch points satisfies b ≤ 2n − 2.

Another corollary is:

Corollary III.4. Let T be a small F_{n}-tree. The stabilizer of any x ∈ T has rank at most n.

In the case of a free action, i(x) + 2 is the number of components of T \ {x}, so that Theorem III.2 specializes to Jiang’s theorem [Ji].

It is worth pointing out the analogy with actions of surface groups. Suppose T is an R-tree with a minimal small action of π_{1}Σ, where Σ is a closed surface. By Skora’s theorem [Sk 1], T is dual to a measured foliation F on Σ. Branch points of T come from singularities of F and the Euler-Poincaré formula for line fields on surfaces gives the equality i(T) = −2χ(Σ) (see Part III).

In the case of F_{n}, equality in Theorem III.2 holds if and only if the action is geometric (compare [Du]). Roughly speaking, geometric means that the action is dual to a measured foliation on a finite 2-complex (see Part II for a discussion). For instance a minimal simplicial F_{n}-action is geometric if and only if every edge stabilizer is finitely generated.

There is a close connection between branch points and rank. This is best seen on geometric F_{n}-actions (not necessarily small). Let L be the subgroup of R generated by the values of the length function ℓ(g) = min_{x∈T} d(x, gx).

For a geometric F_{n}-action, the group L is finitely generated. Its rank r is called the rank of the action (or of the length function). Equivalently T may be viewed as the completion of a Λ-tree, with Λ ⊂ R a subgroup of rank r (see [Sh 2, §1.3.1]).

By studying the 2-group L/2L, we show (Corollary IV.3) the inequality

r ≤ b + n − 1

valid for any geometric minimal F_{n}-action without inversions (the number b of orbits of branch points is always finite). In particular we have r ≤ 3n − 3 for a geometric very small action.

If the action is not geometric, the group L needs not be finitely generated (this may happen for free actions, see Example II.7). Instead of rank we use Q-rank: the dimension of the Q-vector space generated by L. Actions with low Q-rank have been studied extensively ([GS], [GSS]).

Theorem IV.4. Let T be a very small minimal F_{n}-tree. The Q-rank of the action satisfies r_{Q} ≤ 3n − 3. Equality may hold only if the action is free simplicial.

Given a finitely generated group G and an integer k, the space of length functions on G with Q-rank ≤ k has topological dimension at most k (Proposition V.1). We then get:

Theorem V.2. The boundary of Culler-Vogtmann’s outer space Y_{n} has dimension 3n − 5.

This improves the result dim \overline{Y_{n}} = 3n − 4 by Bestvina-Feighn [BF 3].

Theorem IV.4 also implies:

Corollary IV.5. Let T be a very small F_{n}-tree with length function ℓ. Suppose ℓ ∘ α = λℓ with α ∈ Aut(F_{n}) and λ ∈ R^{+}. Then λ is algebraic, of degree bounded by 3n − 4. If T is geometric, then λ is an algebraic unit.

Such an ℓ represents a fixed point for the action of α on \overline{Y_{n}} (compare [Lu]). In Example II.7 we use a construction by Bestvina-Handel to get an example with λ not an unit. The corresponding action is free and does not have finite rank.

The theorems mentioned above are proved in Parts III, IV, V. Parts I and II may be viewed as preliminary.

First recall the following construction due to Rips (see [GLP 1]). Let T be a minimal F_{n}-tree, and K ⊂ T a finite subtree (i.e. a subtree homeomorphic to a finite simplicial complex). If K is large enough, the action of each generator g_{1}, …, g_{n} of F_{n} defines a partial isometry φ_{i}: g_{i}^{−1}K ∩ K → K ∩ g_{i}K between nonempty closed subtrees of K.

In Part I we show how to associate a canonical geometric F_{n}-tree T_{K} to a system K consisting of a finite metric tree K and n partial isometries φ_{i}: A_{i} → B_{i} between closed subtrees of K (Theorem I.1). Similar constructions are known (see e.g. [GLP 2]), but they often require an additional hypothesis to ensure that a certain space is Hausdorff.

In our particular setting this problem does not exist. One consequence, used in the proof of Theorem III.2, is that orbits of branch points of T_{K} are created only by vertices of the finite trees K and A_{i} (i = 1, …, n). Another consequence, derived in Part V, is a simple proof of the following result announced by Skora [Sk 3]:

Theorem V.4. Every action of F_{n} may be approximated by simplicial actions.

Returning to T as above, we associate an F_{n}-tree T_{K} to every finite subtree K ⊂ T. As K grows bigger, these trees approximate T. We define T to be geometric if T equals T_{K} for some K (see Part II for equivalent definitions).

If T is not geometric, it is the strong limit (in the sense of [GS]) of a sequence of geometric actions. This allows us to prove Theorems III.2 and IV.4 first for geometric actions, and then to “pass to the limit”.

In Part II we give examples of geometric and non-geometric actions. In particular we take advantage of the non-completeness of certain minimal F_{n}-trees to construct a lot of non-geometric actions by taking “free products” of actions using a basepoint not in the tree but in its completion (Example II.6).

Acknowledgements. We wish to thank M. Bestvina, M. Dunwoody, F. Paulin, K. Vogtmann for very useful conversations, and the referee for suggesting improvements.

I. THE R-TREE ASSOCIATED TO A SYSTEM OF ISOMETRIES.

Let G be a group. A G-tree is an R-tree T equipped with a left isometric action of G. Two G-trees are considered equal if they are equivariantly isometric.

A finite tree will be an R-tree homeomorphic to a finite simplicial complex. A subtree of an R-tree is a finite tree if and only if it is the convex hull of a finite subset.

A map j: T → T′ between R-trees is a morphism if every segment in T may be written as a finite union of subsegments, each of which is mapped isometrically into T′. If j is an equivariant morphism between G-trees, with length functions ℓ and ℓ′, then ℓ ≥ ℓ′ since j does not increase distances.

We let F_{n} be the free group on n generators g_{1}, …, g_{n}. We write |g| for the length of g ∈ F_{n} relative to this generating set.

We consider systems K consisting of a finite tree K and n isometries φ_{i}: A_{i} → B_{i} between closed nonempty subtrees of K. We let S be the (finite) set consisting of all vertices of the trees K, A_{i}, B_{i} (1 ≤ i ≤ n).

For example, take K to be a finite subtree in an F_{n}-tree T, with K ∩g_{i}K ≠ ∅ for i = 1, …, n. Then define φ_{i} as the restriction of the action of g_{i} to A_{i} = g_{i}^{−1}K ∩K.

Theorem I.1. Let K be as above. There exists a unique F_{n}-tree T_{K} such that: (1) T_{K} contains K (as an isometrically embedded subtree).

(2) if x ∈ A_{i}, then g_{i}x = φ_{i}(x).

(3) every orbit of the action meets K, indeed every segment of T_{K} is contained in a finite union of images wK, w ∈ F_{n}.

(4) if T′ is another F_{n}-tree satisfying (1) and (2), there exists a unique equivariant morphism j: T_{K} → T′ such that j(x) = x for x ∈ K.

Remark I.2. If j is as in (4), it is surjective if and only if T′ satisfies (3).

Remark I.3. Before proving Theorem I.1, we give a geometric description of T_{K}. Let Γ be the Cayley graph of F_{n} relative to g_{1}, …, g_{n}. We construct a foliated 2-complex Σ sitting above Γ, as follows. Place a copy K(g) of K above each vertex g of Γ. Above each edge g − gg_{i}, place a strip A_{i} × [0, 1] foliated by {∗} × [0, 1]. Then glue A_{i} ×{1} to K(gg_{i}) using the inclusion of A_{i} into K, and glue A_{i} ×{0} to the subtree of K(g) corresponding to B_{i}, using φ_{i} (i.e. identify (x, 0) ∈ A_{i} × [0, 1] to φ_{i}(x) ∈ K(g)). The tree T_{K} is the space of leaves of this simply connected foliated 2-complex Σ. The action of F_{n} on T_{K} is induced by the natural action of F_{n} on Γ.

Proof of theorem I.1.

Recall that a pseudodistance on a set X is a symmetric map δ : X × X → R^{+} satisfying the triangle inequality, with δ(x, x) = 0 ∀x. The relation “δ(x, y) = 0” is a (possibly nontrivial) equivalence relation R on X, and δ induces a genuine distance d on the quotient set Y = X/R. We call (Y, d) the metric space associated to (X, δ).

Now suppose T′ is an F_{n}-tree satisfying (1) and (2). Write d′ and d_{K} for distance in T′ and K respectively. Let δ′ be the pseudodistance on K ×F_{n} defined by δ′((x, g), (y, h)) = d′(gx, hy).

A simple computation, based on (1) and (2), shows the inequality

δ′((x, g), (y, h)) ≤ inf{d_{K}(x, x_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(x_{p}), x_{p−1}) + …

+ d_{K}(φ_{i_{2}}^{ε_{2}}(x_{2}), x_{1}) + d_{K}(φ_{i_{1}}^{ε_{1}}(x_{1}), y)} (∗)

where the infimum is taken over all words g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}} representing h^{−1}g (with ε_{j} = ±1) and all points x_{j} in the domain of φ_{i_{j}}^{ε_{j}}.

Indeed we write:

δ′((x, g), (y, h)) = d′(gx, hy)

= d′(h^{−1}gx, y)

= d′(g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}x, y)

\begin{matrix}≤ d′(g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}x, & g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}x_{p}) + d′(g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}x_{p}, g_{i_{1}}^{ε_{1}} … g_{i_{p−1}}^{ε_{p−1}}x_{p−1}) \\ + ⋯ + d′(g_{i_{1}}^{ε_{1}}g_{i_{2}}^{ε_{2}}x_{2}, g_{i_{1}}^{ε_{1}}x_{1}) + d′(g_{i_{1}}^{ε_{1}}x_{1}, y) \end{matrix}

\begin{matrix}≤ d_{K}(x, x_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(x_{p}), x & _{p−1}) + … \\ + d_{K}(φ_{i_{2}}^{ε_{2}}(x_{2}), x_{1}) + d_{K}(φ_{i_{1}}^{ε_{1}}(x_{1}), y). \end{matrix}

With this as a motivation, define δ((x, g), (y, h)) as the infimum in the right hand side of the above inequality (∗). This gives a pseudodistance on K × F_{n}. It induces d_{K} on each K × {g}, and it is invariant under the natural action of F_{n} given by h(x, g) = (x, hg).

It is important to note that the infimum is always achieved: we need only consider the reduced word g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}} representing h^{−1}g, and then the infimum is taken over a fixed number of points x_{j} varying in compact sets.

More explicitly, let z_{p} be the point in the domain of φ_{i_{p}}^{ε_{p}} closest to x, let z_{p−1} be the point in the domain of φ_{i_{p−1}}^{ε_{p−1}} closest to φ_{i_{p}}^{ε_{p}}(z_{p}), and so on. Then:

d_{K}(x, x_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(x_{p}), x_{p−1})
= d_{K}(x, z_{p}) + d_{K}(z_{p}, x_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(x_{p}), x_{p−1})

= d_{K}(x, z_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(z_{p}), φ_{i_{p}}^{ε_{p}}(x_{p})) + d_{K}(φ_{i_{p}}^{ε_{p}}(x_{p}), x_{p−1})

≥ d_{K}(x, z_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(z_{p}), x_{p−1})

and induction on p = |h^{−1}g| yields

δ((x, g), (y, h)) = d_{K}(x, z_{p}) + d_{K}(φ_{i_{p}}^{ε_{p}}(z_{p}), z_{p−1}) + ⋯ + d_{K}(φ_{i_{1}}^{ε_{1}}(z_{1}), y). (∗∗)

We claim that the metric space T_{K} associated to (K ×F_{n}, δ) is an R-tree. Since T_{K} is connected (because A_{i} × {gg_{i}} and B_{i} × {g} have the same image in T_{K}), it suffices by [AB, Theorem 3.17] to show that any 4 points u_{i} = (x_{i}, h_{i}) satisfy the 0-hyperbolicity inequality:

δ(u_{1}, u_{2}) + δ(u_{3}, u_{4}) ≤ max{δ(u_{1}, u_{3}) + δ(u_{2}, u_{4}), δ(u_{1}, u_{4}) + δ(u_{2}, u_{3})}.

This is clear if the elements h_{1}, h_{2}, h_{3}, h_{4} are equal, since K is a tree. In general, we consider them as 4 points in a simplicial tree, namely the Cayley graph Γ of F_{n} relative to g_{1}, …, g_{n}. Let Γ_{0} be the finite subtree they span. Assume that some terminal vertex of Γ_{0}, say h_{1}, is distinct from the other three elements h_{2}, h_{3}, h_{4}. Then the reduced words representing h_{1}, h_{2}^{−1}h_{1}, h_{3}^{−1}h_{1}, h_{4}^{−1}h_{1} all end with the same letter, say g_{1}. Let z be the point in A_{1} closest to x_{1}. We have δ(u_{1}, u_{k}) = d_{K}(x_{1}, z) + δ((φ_{1}(z), h_{1}g_{1}^{−1}), u_{k}) for k = 2, 3, 4, and 0-hyperbolicity follows by induction on the size of Γ_{0}. We leave to the reader the remaining case, when h_{1}, h_{2}, h_{3}, h_{4} are equal in pairs.

The R-tree T_{K} obviously satisfies (1) and (2), with K embedded in T_{K} as K × {1}. Furthermore K meets every orbit.

Since K ∩ g_{i}K ≠ ∅ for i = 1, …, n, any segment in T_{K} joining a point of gK to a point of hK, with hg^{−1} = g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}, may be covered by a finite union of images wK, namely gK, (g_{i_{p}}^{ε_{p}}g)K, (g_{i_{p−1}}^{ε_{p−1}}g_{i_{p}}^{ε_{p}}g)K, …, (g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}g)K. Applied to an arbitrary F_{n}-tree T′ satisfying (1) and (2), this argument shows that the union of all orbits meeting K is a subtree (i.e. it is connected). This means that in proving (4) we may assume that T′ also satisfies (3).

Define δ′ on K × F_{n} as in the beginning of the proof. The map (x, g) ↦ gx identifies T′ with the metric space associated to (K × F_{n}, δ′) (while T_{K} is associated to (K×F_{n}, δ)). Since δ′ ≤ δ, the identity of K×F_{n} induces a continuous equivariant map j: T_{K} → T′. This j induces the identity on K and is a morphism because any segment in T_{K} is contained in a finite union of images of K. Finally, uniqueness of T_{K} is easy to check using (4).

Since the infimum defining δ is always achieved, we have the following facts about T_{K}:

Proposition I.4.

(1) two points (x, g) and (y, h) in K × F_{n} define the same point in T_{K} if and only if one can write y = φ_{i_{1}}^{ε_{1}} … φ_{i_{p}}^{ε_{p}}(x) with g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}} = h^{−1}g.

(2) given x, y ∈ K and g ∈ F_{n}, one has y = gx if and only if one can write y = φ_{i_{1}}^{ε_{1}} … φ_{i_{p}}^{ε_{p}}(x) with g = g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}.

(3) if γ ∈ F_{n} is represented by a cyclically reduced word g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}, then

ℓ(γ) = min_{x_{j}∈domφ_{i_{j}}^{ε_{j}}}{d_{K}(x_{p}, φ_{i_{1}}^{ε_{1}}(x_{1})) + d_{K}(x_{1}, φ_{i_{2}}^{ε_{2}}(x_{2})) + ⋯ + d_{K}(x_{p−1}, φ_{i_{p}}^{ε_{p}}(x_{p}))}.

Remark. In the situation of Assertion 2, note that all points g_{i_{j}}^{ε_{j}} … g_{i_{p}}^{ε_{p}}x (1 ≤ j ≤ p) belong to K.

We now prove a few other properties of T_{K}.

Proposition I.5. If γ ∈ F_{n} is represented by a cyclically reduced word, then its fixed point set Fixγ ⊂ T_{K} is contained in K.

Proof. Let a ∈ T_{K} be a fixed point of γ. Choose a representative (x, g) ∈ K × F_{n} of a with the length |g| minimal. We shall identify g and the reduced word representing it. We assume |g| > 0, and we argue towards a contradiction.

Since (x, g) and (x, γg) both represent a, Proposition I.4 (Assertion 1) lets us write x = φ_{i_{1}}^{ε_{1}} … φ_{i_{p}}^{ε_{p}}(x) with g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}} = g^{−1}γg (and g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}} reduced). Now (φ_{i_{1}}^{−ε_{1}}(x), gg_{i_{1}}^{ε_{1}}) and (φ_{i_{p}}^{ε_{p}}(x), gg_{i_{p}}^{−ε_{p}}) also represent a, so that g cannot end with g_{i_{1}}^{−ε_{1}} or g_{i_{p}}^{ε_{p}} (by minimality of |g|). It follows that gg_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}g^{−1} is the reduced word representing γ. This means that γ cannot be represented by a cyclically reduced word.

If γ ≠ 1 is not cyclically reduced, then Fixγ is contained in some hK. We then get:

Corollary I.6. For any γ ≠ 1 in F_{n}, the set Fixγ ⊂ T_{K} is compact. If j: T_{K} → T′ is a morphism as in Theorem I.1, the restriction of j to Fixγ is an isometry.

Corollary I.7. Suppose the action of F_{n} on T_{K} has no global fixed point. Then its length function is not abelian (i.e. it is not the absolute value of a homomorphism from F_{n} to R).

Proof. Otherwise, commutators would have non-compact fixed point sets (see e.g. [CM, 2.2 and 2.3]).

Recall that S is the finite set consisting of all vertices of the trees K, A_{i}, B_{i} (1 ≤ i ≤ n).

Proposition I.8. If x ∈ T_{K} is a branch point, its orbit contains a point of S. The action of the isotropy subgroup Stab(x) ⊂ F_{n} on the set of directions π_{0}(T_{K} \ {x}) has only finitely many orbits.

Proof. We start with a general argument. Suppose [x, x′] ⊂ T_{K} is a segment with [x, x′] ∩ K = {x}. Some subsegment [x, x_{1}] is contained in some wK. We choose x_{1} ≠ x and w so that p = |w| is minimal. By Proposition I.4 (Assertion 2) we can write x = φ_{i_{1}}^{ε_{1}} … φ_{i_{p}}^{ε_{p}}(y), with y ∈ K and w = g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}. Since [x, x_{1}] ∩ K = {x}, minimality of p implies that the segment [y, y_{1}] = w^{−1}([x, x_{1}]) ⊂ K meets the domain of φ_{i_{p}}^{ε_{p}} only at y. In particular y ∈ S.

This argument shows that the orbit of any branch point x ∈ T_{K} meets S: since K meets every orbit we may assume x ∈ K, and if x is not a vertex of K then there is a segment [x, x′] as above. The argument also implies the second assertion since the number of possible points y ∈ S, and possible germs of segments [y, y_{1}] ⊂ K, is finite.

Corollary I.9. There are only finitely many orbits of branch points in T_{K}.

Remark. The number of orbits of branch points may be bounded in terms of n and the complexity of K. Our goal for very small actions will be to find a bound involving only n.

Corollary I.9 implies that the action on T_{K} is a J-action in the sense of [Le 3]. It follows that the closure of any orbit is a discrete union of closed subtrees. If no orbit is discrete, then every orbit is dense.

We shall use the following fact:

Proposition I.10. Suppose F_{n} acts on T_{K} with every orbit dense. If the action is small, then every edge stabilizer is trivial.

This is well-known (Rips, [BF 3, Remark 1.9]), but we sketch a proof. It is based on a theorem by Imanishi.

Proof.

If the result is false, let E be an edge with stabilizer Z. By shortening E and applying elements of F_{n}, we may assume that every subarc of E has the same stabilizer, and a generator g of Stab(E) is represented by a cyclically reduced word g_{i_{1}}^{ε_{1}} … g_{i_{p}}^{ε_{p}}. Note that E ⊂ K by Proposition I.5.

Choose x ∈ E^{∘} such that the orbit F_{n}(x) contains no point of S. Observe that F_{n}(x) meets K in an infinite set: otherwise F_{n}(x) would be discrete. Imanishi’s theorem (see [GLP 1, Theorem 3.1]) then implies that F_{n}(x) ∩ K accumulates on x. [Theorem 3.1 of [GLP 1] is stated for systems of isometries on a multi-interval, but it also holds on a finite tree]

Consider h ∈ F_{n} such that hx ≠ x belongs to E^{∘}. Then hgh^{−1} stabilizes some neighborhood of hx in E, so that hgh^{−1} is a power of g. It follows that h commutes with g. Since g is cyclically reduced, this leads to a contradiction for hx closer to x than any φ_{i_{j}}^{ε_{j}} … φ_{i_{p}}^{ε_{p}}(x), j = 2, …, p.

Recall that an F_{n}-tree with no global fixed point contains a unique minimal invariant subtree, the union of all translation axes (see [CM]). The following fact will be used in Example II.6, but not elsewhere.

Proposition I.11. Suppose the action of F_{n} on T_{K} has no global fixed point. Then the minimal invariant subtree T_{min} is closed in T_{K}.

Proof. Assume T_{min} is not closed. Then there is a segment [x, y] with [x, y]∩T_{min} = (x, y]. Changing y and applying an element of F_{n}, we may assume [x, y] ⊂ K. We thus see that the tree K′ = T_{min} ∩ K is not closed in K.

It has finitely many limit points x_{1}, …, x_{k}. Let K′′ be the tree obtained from K′ by removing open segments of equal length (x_{1}, y_{1}), …, (x_{k}, y_{k}) disjoint from S. Since T_{min} is connected, we have K′ ∩ A_{i} ≠ ∅ for each i. The same is then true for K′′. This implies that the union of all orbits meeting K′′ is a subtree T′′. By Proposition I.4 (Assertion 2) the intersection of T′′ with K′ consists only of K′′ since no φ_{i}^{ε_{i}} can send a point of some (x_{j}, y_{j}) into K′. We thus get an invariant subtree properly contained in T_{min}, a contradiction.

Remark. The action of F_{n} on T_{min} is the action associated to K′, φ_{1}|K′, …, φ_{n}|K′.

Corollary I.12. Suppose the subgroup F_{p} ⊂ F_{n} generated by g_{1}, …, g_{p} acts with no global fixed point. Then its minimal invariant subtree T_{min}(F_{p}) is closed in T_{K}.

Proof. The union of all F_{p}-orbits meeting K is a subtree T (F_{p}), and the action of F_{p} on T (F_{p}) is the action associated to (K, φ_{1}, …, φ_{p}). The set T_{min}(F_{p}) ∩ K is closed in K (by Proposition I.11), hence also in T_{K} (by an argument given above).

II. GEOMETRIC AND NON-GEOMETRIC ACTIONS.

Let T be a minimal F_{n}-tree, with length function ℓ. Let K ⊂ T be a finite subtree such that K ∩ g_{i}K ≠ ∅ (i = 1, …, n). We consider the system K = (K, (φ_{i})_{i=1,...,n}), with φ_{i} the restriction of the action of g_{i} to g_{i}^{−1}K ∩K (if T = T_{K}, this new K equals the original K because g_{i}^{−1}K ∩ K = A_{i} by Assertion 2 of Proposition I.4: notation is consistent).

Theorem I.1 associates to K an F_{n}-tree T_{K}, with a surjective morphism j_{K}: T_{K} → T. We shall usually write T_{K} instead of T_{K}, and we denote by ℓ_{K} the length function of T_{K}. Recall that ℓ_{K} ≥ ℓ and ℓ_{K} is not abelian (Corollary I.7).

If the action on T is free (resp. small, resp. very small), so is the action on T_{K}: this is clear for free and small actions, and it follows from Corollary I.6 for very small actions.

The tree T_{K} is not necessarily minimal, but we can find arbitrarily large subtrees K with T_{K} minimal, as follows. Fix x_{0} ∈ T. It belongs to some translation axis A_{γ} (see [CM]). Choose an integer p ≥ |γ|, and define K_{p} as the convex hull of the set {gx_{0}; |g| ≤ p} (note that by minimality T is the increasing union of the subtrees K_{p}). Since p ≥ |γ|, all images of x_{0} by terminal subwords of γ belong to K_{p} and it follows that the distance between x_{0} and γx_{0} is the same in T_{K_{p}} as in T. The point x_{0} thus belongs to the axis of γ in T_{K_{p}}. Being the convex hull of the orbit of x_{0}, the F_{n}-tree T_{K_{p}} is minimal.

Now consider two finite subtrees K, K′ of T, with K ⊂ K′. Theorem I.1 provides an equivariant morphism j_{K,K′}: T_{K} → T_{K′}, so that the trees T_{K} form a direct system of F_{n}-trees.

We now prove the well-known fact that this direct system converges strongly towards T in the sense of [GS]. This amounts to showing that, given a segment I in some T_{K}, there exists K′ ⊃ K such that the set j_{K,K′}(I) ⊂ T_{K′} is mapped isometrically into T by j_{K′}. Choose finitely many elements h_{j} ⊂ F_{n} such that I is covered by the trees h_{j}K. Letting m = max |h_{j}|, take any K′ containing all images of K by words of length ≤ m.

To be more concrete, T is the strong limit of the sequence of minimal trees T_{K_{p}} constructed above. The fact that the limit is strong is often used in the following way. Any finite subtree A ⊂ T may be lifted isometrically to T_{K_{p}} for p large: there exists a subtree A^{p} ⊂ T_{K_{p}} such that the restriction of j_{K_{p}} : T_{K_{p}} → T is an isometry. Furthermore, given g ∈ F_{n} and lifts A^{p}, A′^{p} of A and gA respectively, there exists q ≥ p such that A′^{q} = gA^{q}, where A^{q} and A′^{q} denote the images of A^{p} and A′^{p} in T_{K_{q}}. In particular ℓ_{K_{q}}(g) = ℓ(g) for q large.

Instead of viewing T as the strong limit of a sequence T_{K_{p}}, we can also choose an increasing continuous family K(t) (t ∈ R^{+}), with T = ∪K(t), and view T as the strong limit of the system T_{K(t)}. The following properties then hold.

Fix g ∈ F_{n}, and consider the function σ_{g}: t ↦ ℓ_{K(t)}(g). It is non-increasing, and it is constant for t larger than some t_{0} (depending on g). Furthermore σ_{g} is continuous: by Proposition I.4 (Assertion 3) we can bound |σ_{g}(t_{1}) − σ_{g}(t_{2})| by |g| times the Hausdorff distance between K(t_{1}) and K(t_{2}).

Now we prove:

Proposition II.1. Let T be a minimal F_{n}-tree. The following conditions are equivalent:

(1) There exists K = (K, φ_{1}, …, φ_{n}) such that T = T_{K}.

(2) There exists a finite subtree K ⊂ T such that T = T_{K} (i.e. j_{K}: T_{K} → T is an isometry).

(2’) There exists a finite subtree K ⊂ T such that ℓ_{K′} = ℓ for every K′ ⊂ T

containing K.

(3) T can only be a strong limit in a trivial way (if T is the strong limit of a sequence of F_{n}-morphisms f_{p}: T_{p} → T_{p+1} between minimal trees, then f_{p} is an isometry for p large).

Proof.

2 ⟹ 1 by definition.

1 ⟹ 2 because (T_{K})_{K} = T_{K} (see the above remark about consistency of notation).

2 ⟹ 2′ because ℓ = ℓ_{K} and ℓ_{K} ≥ ℓ_{K′} ≥ ℓ.

2′ ⟹ 2: Take K′ containing K such that T_{K′} is minimal. Then T_{K′} and T are equal because they are minimal trees with the same, non-abelian, length function ([AB], [CM]).

3 ⟹ 2 because T = T_{K_{p}} for p large.

To prove 1 ⟹ 3, suppose that T = T_{K} is the strong limit of a sequence f_{p}. For p large enough we may lift K isometrically to a subtree K^{p} of T_{p} (see above). For i = 1, …, n, let A_{i}^{p} and B_{i}^{p} be the subtrees of K^{p} corresponding to A_{i} and B_{i}. Since g_{i}A_{i} = B_{i} we may take p even larger so as to ensure g_{i}A_{i}^{p} = B_{i}^{p}, and Theorem I.1 yields a morphism j: T → T_{p}. It follows that the morphism from T_{p} to the limit tree T is an isometry, and the strong limit is trivial.

Definition. A minimal action of F_{n} is geometric if it satisfies conditions 1-3 above. Using condition 3, we see that being geometric or not does not depend on the particular set of generators g_{1}, …, g_{n}.

Example II.2. We have seen that every minimal action of F_{n} is the strong limit of a sequence of geometric minimal actions.

Example II.3. The non-simplicial free F_{3}-actions constructed in [Le 2] are geometric.

Example II.4. An F_{n}-action with an abelian length function is not geometric by Corollary I.7 (compare [Le 4]).

Example II.5. It may be shown that a minimal simplicial F_{n}-action is geometric if and only if every edge stabilizer has finite rank. In particular, small simplicial actions are geometric.

Example II.6: non-geometric free products of actions

Consider finitely generated free groups G_{1}, G_{2} acting non-trivially on R-trees T_{1} and T_{2}. Fix basepoints p_{i} ∈ T_{i}. One can combine these two actions ([Sk 2], [CL]), obtaining an R-tree T with an action of G_{1} ∗ G_{2}. If the actions on T_{i} are minimal (resp. free), so is the action on T. More generally, the action on T is minimal as soon as no proper G_{i}-invariant subtree of T_{i} contains p_{i}.

Now let T_{1} be a minimal G_{1}-tree. Assume that branch points are dense in T_{1} (this happens for instance for the free F_{3}-actions of [Le 2], or for certain very small F_{2}-actions). Then segments are nowhere dense closed subsets, and by Baire’s theorem T_{1} is not complete as a metric space since it is a countable union of segments.

Choose a point p_{1} in the completion \overline{T_{1}} but not in T_{1}, and let T′_{1} ⊂ \overline{T_{1}} be the smallest G_{1}-invariant subtree containing p_{1}. Combine the G_{1}-tree T′_{1} with some minimal G_{2}-tree T_{2} (e.g. G_{2} = Z, T_{2} = R). The resulting (G_{1} ∗ G_{2})-tree T is minimal, but by Corollary I.12 it is not geometric since T_{1} is not closed in T.

Example II.7: a free F_{3}-action with L infinitely generated

Bestvina-Handel have shown how iterating an automorphism of F_{n} may lead to a non-simplicial free F_{n}-action. An explicit example is worked out in [Sh 2]. It is not geometric because it is a nontrivial strong limit (compare [BF 3]). We give an example where iterating an automorphism of F_{3} leads to a free action such that the values of the length function do not lie in any finitely generated subgroup of R.

Let α be the automorphism of F_{3} given by α(a) = ab^{−1}, α(b) = bac^{−1}, α(c) = ca^{−3}. Let λ > 1 be the largest eigenvalue of the associated matrix

A = ( \begin{matrix}1 & 1 & 0 \\ 1 & 1 & 1 \\ 3 & 0 & 1 \end{matrix} )

and let (u, v, w) be a positive eigenvector.

View F_{3} as the fundamental group of a wedge of 3 circles of respective lengths u, v, w, and let ℓ: F_{3} → R^{+} be the corresponding length function (associated to the action of F_{3} on the universal covering).

Since ℓ(αh) ≤ λℓ(h) for every h ∈ F_{n}, each sequence

ℓ_{p}(g) = λ^{−p}ℓ(α^{p}g)

is non-increasing. Taking its limit as p → +∞, we get a function ℓ_{∞}: F_{3} → R^{+} which is the length function of a very small action (provided it is not identically 0).

Our discussion so far holds for any automorphism of F_{3} (or even of F_{n}), as long as the matrix A has a positive eigenvector. We now use the special form of α.

First of all, arguing as in [Sh 2], one shows that each sequence ℓ_{p}(g) is eventually constant, so that ℓ_{∞}(g) is positive for every nontrivial g ∈ F_{n}. Thus ℓ_{∞} is the length function of a free action.

Now the key feature of our example is that λ is not an algebraic unit, because the determinant of A is 3 (it is always an odd integer because A is invertible mod 2). This implies that Z[λ, λ^{−1}] is not a finitely generated subgroup of R. Since ℓ_{∞} satisfies the relation

ℓ_{∞}(α^{±1}g) = λ^{±1}ℓ_{∞}(g),

the subgroup L ⊂ R generated by the values of ℓ_{∞} is a Z[λ, λ^{−1}]-module and therefore is not a finitely generated group.

Remark II.8.

- It is easy to check that α^{−2} is a positive automorphism. On the other hand α could not be positive, since det A = ±1 if α is positive.

- One can show that the F_{3}-action just constructed has only one orbit of branch points. These branch points have index 1 (i.e. T \ {x} has 3 components).

- The second author has proved that very small F_{n}-actions with Q-rank 3n − 4 have finite (Z-)rank.

III. COUNTING BRANCH POINTS.

Let T be a minimal small F_{n}-tree. Given x ∈ T, a direction d from x is a component of T \{x}, or equivalently a germ of edges issuing from x. The isotropy subgroup Stab(x) ⊂ F_{n} acts on the set of directions from x. The stabilizer Stab(d) of a direction d is either trivial or infinite cyclic.

Let v_{1}(x) be the (presumably infinite) number of Stab(x)-orbits of directions from x with trivial stabilizer. We define the index

i(x) = 2 rk Stab(x) + v_{1}(x) − 2.

Theorem III.2 will imply that i(x) is finite. If Stab(x) is trivial, then i(x) + 2 is the number of components of T \ {x}.

This definition may be motivated by the analogy with surface groups mentioned in the introduction. Suppose F is a measured foliation on a closed surface Σ, whose singularities are k_{s}-prong saddles (k_{s} ≥ 3). Then ∑_{s}(k_{s} − 2) = −2χ(Σ) by the Euler-Poincaré formula [FLP, p. 75]. A branch point x in the π_{1}Σ-tree associated to F corresponds to a set A of saddles linked by saddle connections. Setting i(x) = ∑_{s∈A}(k_{s} − 2) leads to the formula above, since v_{1}(x) is the number of infinite separatrices issuing from saddles in A while Stab(x) is isomorphic to the fundamental group of the 1-complex whose edges are the saddle connections.

Proposition III.1. The index i(x) is always non-negative. If i(x) > 0, then x is a branch point. Conversely, if the action is very small, then every branch point has index ≥ 1.

Proof.

We fix x ∈ T, and we distinguish three cases according to the rank of Stab(x). If Stab(x) has rank ≥ 2, then i(x) ≥ 2. Since the action of Stab(x) on the set

of directions has an infinite orbit, x is a branch point.

If Stab(x) is trivial, then i(x) = v_{1}(x) − 2, with v_{1}(x) equal to the number of components of T \ {x}. Minimality of the action implies v_{1}(x) ≥ 2. We thus have i(x) ≥ 0, and i(x) > 0 if and only if x is a branch point.

If Stab(x) ≃ Z, then i(x) = v_{1}(x) is non-negative. If i(x) > 0, we deduce that x is a branch point as in the first case. Now we assume that i(x) = v_{1}(x) is 0 and the action is very small, and we prove x is not a branch point.

Consider a direction from x. The inclusion from its stabilizer into Stab(x) is an isomorphism because there are no obtrusive powers. This means that every element of Stab(x) acts on π_{0}(T \ {x}) as the identity. By the no-triod condition, there cannot be 3 distinct directions from x, so that x is not a branch point.

Remark. The proof shows that a branch point x has index 0 if and only if Stab(x) ≃ Z and v_{1}(x) = 0.

Clearly i(x) = i(x′) if x and x′ belong to the same F_{n}-orbit O, and we write i(O) = i(x). We define the total index of T as

i(T) = ∑_{O∈T/F_{n}} i(O).

Theorem III.2. Let T be a minimal small F_{n}-tree.

(1) If T is geometric, then i(T) = 2n − 2.

(2) If T is not geometric, then i(T) < 2n − 2.

Corollary III.3. If T is a minimal very small F_{n}-tree, the number of orbits of branch points is at most 2n − 2.

Corollary III.4. If T is a minimal small F_{n}-tree, the stabilizer of any x ∈ T has rank at most n.

Proof of theorem III.2.

First assume that T = T_{K} is geometric. Given a finite tree H (such as K or A_{i}), and x ∈ H, we denote u_{H}(x) the valence of x in H. Then:

∑_{x∈H}(u_{H}(x) − 2) = −2.  (1)

Fix an F_{n}-orbit O ⊂ T_{K}. The interesting case is when O contains a point of S (since otherwise i(O) = 0 by Propositions I.8 and III.1), but for now O may be arbitrary. We define a “Cayley graph” O_{K} as follows. Vertices of O_{K} are the points of O belonging to K (recall that K meets every orbit). There is an edge labelled g_{i} from z to φ_{i}(z) whenever z ∈ A_{i}. Assertion 2 of Proposition I.4 implies that O_{K} is connected.

We define the weight w(e) of an edge e labelled g_{i} as the valence of its origin z in A_{i}. All but finitely many edges have weight 2.

Next we define a “blown-up” 1-complex O′_{K}. Vertices of O′_{K} will be directions, viewed as germs of edges. If x ∈ K, we shall distinguish between directions from x in K or in T_{K}.

To define O′_{K}, we start from O_{K}, replacing each vertex x of O_{K} by u_{K}(x) vertices representing directions d from x in K, and replacing each edge e by w(e) edges in the obvious way (these edges in O′_{K} carry the same label and orientation as e). Let π be the natural projection from O′_{K} to O_{K}.

Lemma III.5. Fix x ∈ O ∩ K.

(1) The fundamental group of O_{K} is isomorphic to Stab(x).

(2) The set of components O_{1} of O′_{K} is in one-to-one correspondence with the set of orbits under Stab(x) of directions d from x in T_{K}.

(3) The fundamental group of a component O_{1} is isomorphic to the corresponding isotropy subgroup Stab(d), hence to {1} or Z.

Proof.

There is a natural homomorphism ρ: π_{1}(O_{K}, x) → F_{n}, where ρ(γ) is the product of labels of edges crossed by a loop γ (taking orientation into account: write g_{i} if the edge is crossed from z to φ_{i}(z) and g_{i}^{−1} otherwise). Clearly this homomorphism is injective and takes values in Stab(x). By Proposition I.4 (Assertion 2), its image is the whole of Stab(x). This proves the first assertion of the lemma.

Now consider a component O_{1} of O′_{K}. A vertex of O_{1} is a direction d_{0} in K, at a point y ∈ O ∩ K. Applying any g ∈ F_{n} taking y to x, we get a direction d from x in T_{K}. The orbit of d under Stab(x) is independent of the choice of either d_{0} or g, and we obtain a map Φ from components of O′_{K} to orbits of directions from x in T_{K}.

The argument used in the proof of Proposition I.8 shows that Φ is onto. To prove injectivity, suppose that d_{0}, d′_{0} are directions in K that correspond to directions d, d′ in the same Stab(x)-orbit. Then some g ∈ F_{n} maps d_{0} to d′_{0}. Assertion 2 of Proposition I.4 implies that d_{0} and d′_{0} belong to the same component of O′_{K}.

Finally, the proof of Assertion (3) is similar to that of Assertion (1).

Thanks to Lemma III.5 we can now deduce properties of T from combinatorial properties of finite subgraphs of O_{K} and O′_{K}. Let G be a finite connected subgraph of O_{K} containing every vertex belonging to S and every edge of weight ≠ 2 (if there are any). Let G′ ⊂ O′_{K} be the preimage π^{−1}(G).

By Proposition I.8 and Lemma III.5, the 1-complex O′_{K} has finitely many components, each with first Betti number 0 or 1. Enlarging G if necessary, we may assume that π_{1}G′ generates the fundamental group of every component of O′_{K}.

Note that in general the intersection of G′ with a given component of O′_{K} need not be connected. These intersections are connected, however, if π_{1}G generates π_{1}O_{K}, since G then contains any embedded path in O_{K} with endpoints in G. It is true that π_{1}O_{K} is finitely generated, but we do not know it yet.

In any connected finite 1-complex we have the formula:

1 − rk π_{1} = ♯ vertices − ♯ edges.  (2)

Applying it to each component G′_{j} of G′ and summing up we get:

∑_{j}(1 − rk π_{1}G′_{j}) = ∑_{x∈V (G)} u_{K}(x) − ∑_{e∈E(G)} w(e),

denoting V and E the set of vertices and edges respectively.

Subtracting formula (2) applied to G and multiplied by 2, we obtain:

2rk π_{1}G − 2 + ∑_{j}(1 − rk π_{1}G′_{j}) = ∑_{x∈V (G)}(u_{K}(x) − 2) − ∑_{e∈E(G)}(w(e) − 2). (3)

The right hand side is independent of G (only finitely many terms may be nonzero), while every term (1 − rk π_{1}G′_{j}) is non-negative. It follows that rk π_{1}G is bounded, in other words π_{1}O_{K} is finitely generated.

We may then assume that π_{1}G generates π_{1}O_{K}, thereby ensuring that a given component of O′_{K} contains only one G′_{j} (see above). By Lemma III.5 this implies that the left hand side of (3) equals i(O), so that we have proved:

i(O) = ∑_{x∈V (O_{K})}(u_{K}(x) − 2) − ∑_{e∈E(O_{K})}(w(e) − 2).

If O does not meet S, the right hand side is 0 and i(O) = 0. For the other orbits we recall that the weight of an edge labelled g_{i} is the valence of its origin in A_{i}, and we write:

i(O) = ∑_{x∈O∩K}(u_{K}(x) − 2) − ∑_{i=1}^{n} ∑_{x∈O∩A_{i}}(u_{A_{i}}(x) − 2).  (4)

Summing up and using (1) we get the required equation:

∑_{O∈T/F_{n}} i(O) = −2 + 2n.

This completes the proof of Theorem III.2 when T is geometric. From now on we assume that T is not geometric (compare [Du, Theorem 5]).

Choose a base point x_{0} ∈ T, and let K_{p} be the convex hull of {gx_{0}; |g| ≤ p}. Recall from the beginning of Part II that T_{K_{p}} is a sequence of minimal small F_{n}-trees converging strongly to T. For convenience we write T_{p} instead of T_{K_{p}}, and we denote j_{p} the morphism T_{p} → T.

Let x be a branch point of T, and k ≤ i(x) an integer (if we knew that i(x) is finite, we would simply take k = i(x)). We are going to show that, for p large enough, there exists a lift x′ ∈ j_{p}^{−1}(x) with i(x′) ≥ k. This will prove i(T) ≤ 2n−2 (note that lifts x′ and y′ are in distinct orbits if x, y are in distinct orbits).

Choose h_{1}, …, h_{q} ∈ Stab(x) belonging to a free basis, and directions d_{1}, …, d_{r} from x with trivial stabilizers, in distinct Stab(x)-orbits, with 2q + r − 2 = k. Because of strong convergence, it is possible for p large to lift x to x′ ∈ T_{p} in such a way that h_{α} fixes x′ (α = 1, …, q). Similarly we may assume that d_{β} lifts to a direction d′_{β} from x′ in T_{p} (β = 1, …, r). Clearly v_{1}(x′) ≥ r. On the other hand j_{p} induces an injection from Stab(x′) into Stab(x) whose image contains h_{1}, …, h_{q}. Since the subgroup generated by h_{1}, …, h_{q} is a free factor of Stab(x), we get rk Stab(x′) ≥ q and i(x′) ≥ k. This shows the existence of x′, hence the inequality i(T) ≤ 2n − 2.

Now we assume i(T) = 2n − 2 and we argue towards a contradiction (for T non-geometric). We assume that the basepoint x_{0} is a branch point.

Consider B ⊂ T containing one point from each orbit with positive index. For x ∈ B choose a basis h_{1}, …, h_{q} of Stab(x) and directions d_{1}, …, d_{r} as before with 2q + r − 2 = i(x). Choose p so that we can associate x′ ∈ T_{p} as above to each x ∈ B. Also make sure that x_{0} is a branch point of K_{p}.

Since i(T_{p}) = i(T), the orbit of every branch point of T_{p} with positive index contains some x′. Furthermore every direction from x′ with trivial stabilizer is Stab(x′)-congruent to some d′_{β}.

The morphism j_{p} is not an isometry. Thus two distinct germs of edges e_{1}, e_{2} at some y ∈ T_{p} are carried by j_{p} onto the same germ at j_{p}(y). We show that this leads to a contradiction.

If y is a branch point with positive index, previous remarks imply that e_{1} and e_{2} both have nontrivial stabilizer. Since e_{1} and e_{2} get identified in T, the union of their stabilizers generates a cyclic group, so that e_{1} ∪ e_{2} is contained in some Fixγ ⊂ T_{p}. This contradicts Corollary I.6.

If y is a branch point with index 0, then again e_{1} and e_{2} both have nontrivial stabilizer (see the remark after the proof of Proposition III.1), and the argument is the same.

If y is a regular point, then e_{1} ∪e_{2} is contained in some wK_{p}: otherwise y would belong to the orbit of a terminal vertex of K_{p}, hence to the orbit of the branch point x_{0} of T_{p}. We have again reached a contradiction since the restriction of j_{p} to wK_{p} is an isometry.

Remark III.6. The total index of a very small minimal F_{n}-tree is at least 1. As mentioned in Remark II.8, the free F_{3}-tree of Example II.7 has index 1. There exist small F_{2}-actions with total index 0.

IV. BOUNDING THE RANK.

Let G be a finitely generated group. Let T be a G-tree, with length function ℓ. Let L be the subgroup of R generated by the values of ℓ. The Q-rank r_{Q} (of the

action, or of the length function) is the dimension of the Q-vector space L ⊗_{Z} Q generated by L. The Z-rank (or simply rank) r is the rank of the abelian group L. Both ranks may be infinite. If r is finite, then r_{Q} = r and L/2L is isomorphic to (Z/2Z)^{r}. A minimal action is simplicial if and only if it is topologically conjugate to an action with r = 1.

Our main interest will be in very small F_{n}-actions, but for now we only assume that T ≠ R is a minimal G-tree with non-abelian length function. Define Λ as the subgroup of R generated by distances between branch points. It is the smallest subgroup Λ ⊂ R such that T may be viewed as the completion of a Λ-tree (see [Sh 2, §1.3.1], or “base change” in [AB]).

We note the inclusions

2Λ ⊂ L ⊂ Λ,

which imply that we may use Λ instead of L when computing r_{Q} and r.

The second inclusion is obvious since we assume T ≠ R. The first one comes from [AB, Theorem 7.13 (c)]. Here is a proof based on the fact that, in a minimal G-tree with non-abelian length function, every segment is contained in some translation axis (see Lemma 4.3 of [Pa]). Given two branch points a, b, there exists a translation axis A_{α} (resp. A_{β}) containing a (resp. b) and disjoint from the open segment (a, b). The well known formula ℓ(αβ) = ℓ(α) + ℓ(β) + 2d(a, b) (see e.g. [Pa, Proposition 1.6]) then yields 2Λ ⊂ L.

We shall say that g ∈ G acts as an inversion if it interchanges two distinct points of T. We thank the referee for pointing out that the following result applies only to actions without inversions (unless we count centers of inversions as branch points). Of course a very small F_{n}-action has no inversion.

Proposition IV.1. Let T ≠ R be a minimal G-tree with non-abelian length function and no inversion.

(1) Let g_{1}, …, g_{n} be a system of generators for G. The numbers ℓ(g_{1}), …, ℓ(g_{n})

generate L mod 2Λ.

(2) Let (p_{j})_{j∈J} be representatives of G-orbits of branch points. For j_{0} ∈ J, the numbers d(p_{j_{0}}, p_{j}) generate Λ mod L.

Proof. First we prove the following equalities modulo 2Λ:

d(b, b′) + d(b′, b′′) = d(b, b′′)  (5)
d(b, gb) = ℓ(g),  (6)

where b, b′, b′′ are branch points and g ∈ G.

Define c by [b, b′] ∩ [b′, b′′] = [b′, c]. The point c is a branch point (possibly b, b′, or b′′), and (5) follows from the formula

d(b, b′) + d(b′, b′′) = d(b, b′′) + 2d(c, b′).

For (6), we use the formula

d(b, gb) = ℓ(g) + 2d(b, C_{g})

where C_{g} is the characteristic set of g (its fixed point set or its translation axis), see e.g. [CM, 1.3]. If the point of C_{g} closest to b is a branch point, we are done. Otherwise g has a unique fixed point, namely the midpoint m of [b, gb]. Since m is not a branch point, the segment g([m, gb]) = [m, g^{2}b] meets [m, b] in a nondegenerate segment and g is an inversion, a contradiction.

If g_{1}, g_{2} ∈ G we choose an arbitrary branch point b, and using (5) and (6) we write (modulo 2Λ):

ℓ(g_{1}g_{2}) = d(b, g_{1}g_{2}b) = d(b, g_{1}b)+d(g_{1}b, g_{1}g_{2}b) = d(b, g_{1}b)+d(b, g_{2}b) = ℓ(g_{1})+ℓ(g_{2}).

This proves Assertion (1) of the proposition since g ↦ ℓ(g) induces a homomorphism from G onto L/2Λ.

Given two branch points q, r, we write q = gp_{j} and r = hp_{k} with g, h ∈ G and j, k ∈ J. Then (also modulo 2Λ):

d(q, r) = d(gp_{j}, hp_{k})
= d(gp_{j}, gp_{j_{0}}) + d(gp_{j_{0}}, gp_{k}) + d(gp_{k}, hp_{k})
= d(p_{j}, p_{j_{0}}) + d(p_{j_{0}}, p_{k}) + ℓ(g^{−1}h)
= d(p_{j_{0}}, p_{j}) + d(p_{j_{0}}, p_{k}) (mod L).  (7)

This proves the second assertion.

Proposition IV.2.

(1) Geometric F_{n}-actions have finite rank.

(2) Consider a non-geometric F_{n}-tree T as the strong limit of a system T_{K(t)}, as in Part II. If lim inf_{t→+∞} r(T_{K(t)}) is finite, then

r_{Q}(T) ≤ lim inf_{t→+∞} r(T_{K(t)})

and

r_{Q}(T) < lim sup_{t→+∞} r(T_{K(t)}).

Proof.

Let T = T_{K}. It follows from Proposition I.8 and Equation (∗∗) (from the proof of Theorem I.1) that Λ is contained in the subgroup of R generated by distances between points in the finite set S, and Assertion (1) holds.

Recall from Part II that for a given g ∈ F_{n} the function t ↦ ℓ_{K(t)}(g) is continuous, and constant for t large. Thus every finitely generated subgroup of L(T) is contained in L(T_{K(t)}) for t large. This proves the first inequality of Assertion (2).

If the second inequality is false, then r(T_{K(t)}) = r_{Q}(T) for t large. We choose a finite set of elements g_{j} ∈ G such that the numbers ℓ(g_{j}) generate L(T) ⊗_{Z} Q. Since each function t ↦ ℓ_{K(t)}(g_{j}) is constant for t large, we see that the Q-vector space generated by L(T_{K(t)}) is independent of t for t large.

Since ℓ_{K(t)} varies continuously, this means that ℓ_{K(t)} is constant for t large. As ℓ_{K(t)} is not abelian (Corollary I.7), the minimal invariant subtree of T_{K(t)} is independent of t. Therefore T is geometric, a contradiction.

Corollary IV.3. Let T be a geometric minimal F_{n}-tree without inversions. Let b be the number of orbits of branch points. Then r(T) ≤ n + b − 1.

Proof. We know by Proposition IV.2 that the action has finite rank r. The group Λ/2Λ is then isomorphic to (Z/2Z)^{r}. The result follows since Λ/2Λ is generated by n + b − 1 elements by Proposition IV.1 (note that b is finite by Corollary I.9).

We now prove:

Theorem IV.4. Let T be a minimal, very small, F_{n}-tree. Then r_{Q}(T) ≤ 3n − 3. Equality may hold only if the action is free and simplicial.

Proof.

If T is geometric, we have r(T) ≤ 3n − 3 by Corollaries III.3 and IV.3. If T is not geometric, we recall that the geometric trees T_{K(t)} are very small (see Part II), so that r_{Q}(T) < 3n − 3 by Proposition IV.2.

From now on we assume that the action is geometric, but not free simplicial. We know that it has finite rank r, with Λ/2Λ ≃ (Z/2Z)^{r}, and we show r < 3n − 3. This will complete the proof.

We consider several cases.

• If the action is simplicial, it is obtained from a graph of groups Γ. Consider the natural epimorphism ρ from F_{n} to the fundamental group of Γ in the topological sense. Since the action is not free, some vertex group is nontrivial and ρ is not injective. Free groups being hopfian, the rank of π_{1}Γ is strictly inferior to n.

On the other hand, every vertex of Γ is the projection of a branch point of T (because there is no inversion). By Corollary III.3, Γ has at most 2n − 2 vertices. It follows that Γ has strictly less than 3n − 3 edges. Since Λ is generated by the lengths of edges, we have r < 3n − 3.

• Now suppose that every F_{n}-orbit is dense in T. In the previous case, we had r < 3n − 3 because L/2Λ had 2-rank < n. In this case, we prove that Λ/L has 2-rank < 2n − 3, so that Λ/2Λ has 2-rank < 3n − 3.

We write T = T_{K} as in Part II, making sure that every terminal vertex of K is a branch point of T. If there are less than 2n − 2 distinct orbits of branch points in T, we are done by Proposition IV.1 (Assertion 2). If not, let p_{1}, …, p_{2n−2} be representatives of these orbits, chosen to belong to K. Each p_{j} has index 1.

By Proposition I.10, every edge stabilizer is trivial. This means that the generators φ_{1}, …, φ_{n} are independent in the sense of [GLP 1]: a reduced word φ_{i_{1}}^{ε_{1}} … φ_{i_{p}}^{ε_{p}} cannot be equal to the identity on a non-degenerate subinterval of K. Denoting by | | arclength in K, we then have

|K| = ∑_{i=1}^{n} |A_{i}|  (8)

([Le 5, Theorem 2], see also [Le 1, corollaire II.5] and [GLP 1, Part 6]).

Equation (8) is an equality between numbers of the form d(q, r), where qr is an edge of K or A_{i}. We view it as an equation in Λ/L (recall that every vertex of K, hence also of A_{i}, is a branch point of T).

Using Equation (7) from the proof of Proposition IV.1, we may replace each term d(q, r) by a sum d(p_{1}, p_{j})+d(p_{1}, p_{k}). We thus obtain a linear relation between the numbers d(p_{1}, p_{j}), j = 2, …, 2n − 2 (whose coefficients are integers mod 2). We have to check that it is not trivial.

The coefficient of d(p_{1}, p_{j}) in the expansion of |K| (resp. |A_{i}|) has the same parity as the sum ∑ u_{K}(x) (resp. ∑ u_{A_{i}}(x)) taken over vertices of K (resp. A_{i}) belonging to the orbit of p_{j}. Since every p_{j} has index 1, Equation (4) from the proof of Theorem III.2 then yields the nontrivial relation ∑_{j=2}^{2n−2} d(p_{1}, p_{j}) = 0 mod L between the 2n − 3 generators of Λ/L.

• Finally, we simply assume that the action is not simplicial. We recall [Le 3] that T may be obtained as a graph of transitive actions. In particular, there exists a subtree T_{v} ⊂ T such that:

- T_{v} is closed, not equal to a point;

- there exists δ > 0 such that, for g ∈ F_{n}, either gT_{v} = T_{v} (i.e. g ∈ Stab(T_{v})) or the distance between T_{v} and gT_{v} is greater than δ;

- Stab(T_{v}) acts on T_{v} with dense orbits.

Let T′ be the F_{n}-tree obtained by collapsing each gT_{v} to a point. The natural action of F_{n} on T′ is very small. Apply Theorem III.2 to both T and T′. We find that Stab(T_{v}) has some finite rank p and

i(T) − i(T′) = i(T_{v}) − (2p − 2),

where i(T_{v}) is the index of T_{v} viewed as a Stab(T_{v})-tree. The left hand side is non-negative because T is geometric, while the right hand side is non-positive. This implies i(T_{v}) = 2p − 2: the action of Stab(T_{v}) on T_{v} is geometric.

If there are less than 2p−2 distinct Stab(T_{v})-orbits of branch points in T_{v}, then there are less than 2n − 2 distinct F_{n}-orbits in T, and we are done. Otherwise, the analysis of the previous case yields a nontrivial relation in Λ(T_{v})/L(T_{v}), hence also in Λ(T)/L(T).

Corollary IV.5. Let T be a very small F_{n}-tree with length function ℓ. Suppose ℓ ∘ α = λℓ with α ∈ Aut(F_{n}) and λ ∈ R^{+}. Then λ is algebraic, of degree bounded by 3n − 4. If T is geometric, then λ is an algebraic unit.

Proof. If the action on T is free simplicial, then λ = 1. If not, multiplication by λ defines an automorphism of L ⊗ Q, a Q-vector space of dimension ≤ 3n − 4. This implies that λ is algebraic of degree ≤ 3n − 4. If the action is geometric, then λ is a unit because it acts on L, a finitely generated abelian group by Assertion 1 of Proposition IV.2.

V. SPACES OF LENGTH FUNCTIONS.

Let G be a finitely generated group. Let Ω be the set of conjugacy classes in G. Let LF (G) ⊂ (R^{+})^{Ω} be the space of all length functions on G, and P LF (G) the space of projectivized length functions. Recall that P LF (G) is compact [CM]. Also note that the Q-rank of a length function ℓ depends only on its class in P LF (G).

Proposition V.1. Let G be a finitely generated group. Let k ≥ 1 be an integer. The space LF_{≤k}(G) of all length functions with Q-rank ≤ k has dimension ≤ k. The space P LF_{≤k}(G) of all projectivized length functions with Q-rank ≤ k has topological dimension ≤ k − 1.

Proof. Fix k + 1 rationally independent real numbers λ_{0}, …, λ_{k}. For j = 0, …, k, let M_{j} be the space of all x ∈ (R^{+})^{Ω} such that no nonzero coordinate of x is a rational multiple of λ_{j}. Each M_{j} has dimension 0: every x ∈ M_{j} has arbitrarily small neighborhoods with boundary disjoint from M_{j}. Next we observe that

every ℓ ∈ LF_{≤k}(G) belongs to at least one M_{j}: otherwise L ⊗ Q would contain λ_{0}, …, λ_{k}. It follows that LF_{≤k}(G) has dimension ≤ k since it is contained in the union of the 0-dimensional sets M_{j}, j = 0, …, k (see [HW, p. 29]). A similar argument applies to P LF_{≤k}(G).

Theorem V.2. The boundary of Culler-Vogtmann’s outer space Y_{n} has dimension 3n − 5.

Proof. The boundary of Y_{n} consists of (projective classes of length functions of) very small actions of F_{n} which are not free simplicial, so that it has dimension ≤ 3n − 5 by Theorem IV.4 and Proposition V.1. Since it is easy to find in δY_{n} a (3n − 5)-simplex consisting of simplicial actions, we have dim Y_{n} = 3n − 5.

Remark V.3. Let T_{1} be a very small F_{2}-tree with dense orbits (see [CV, §5]). Apply Example II.6, taking T_{2} to be the universal covering of the graph Γ pictured below (for n ≥ 3) and choosing p_{2} in the preimage of q. We get a non-geometric very small F_{n}-tree. Varying the lengths of edges of Γ gives a (3n − 7)-simplex of non-geometric actions in δY_{n} (this application of Example II.6 was suggested by M. Bestvina). Since we may choose p_{1} arbitrarily in \overline{T_{1}} \ T_{1}, which is onedimensional (see the proof of Theorem 2.2.2 in [MNO]), the set of non-geometric actions in δY_{n} has dimension ≥ 3n − 6 for n ≥ 3.

4.71in by 1.32in (Fig1 scaled 1030)

Finally, we sketch a proof of a theorem announced by Skora [Sk 3].

Theorem V.4. Length functions of simplicial actions are dense in LF (F_{n}).

Proof. We need to approximate any ℓ ∈ LF (F_{n}) by simplicial length functions. By Example II.2, we may assume that ℓ comes from a geometric F_{n}-tree T_{K}. The system K consists of a finite tree K and n isometries φ_{i}: A_{i} → B_{i}. We may approximate it by a system K′ such that every distance between vertices of K′, A′_{i}, B′_{i} is rational. The corresponding length function ℓ′ is then simplicial. By Assertion 3 of Proposition I.4, this ℓ′ is an approximation of ℓ: for g ∈ F_{n} cyclically reduced, |ℓ′(g) − ℓ(g)| is bounded by |g| times a constant depending only on K and K′.

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Laboratoire de Topologie et Géométrie, URA CNRS 1408, Université Toulouse III, 31062 Toulouse Cedex, France.

Current address of D. Gaboriau: Unité de Mathématiques CNRS UMR 128, Ecole Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France.

e-mail: gaboriau@umpa.ens-lyon.fr levitt@cict.fr