Bounding the complexity of small actions on ℝ-trees.

Vincent Guirardel

September 6, 2001

1

Abstract

We bound the number of orbits of strongly reduced branch points and the number of orbits directions from a branch point in any small stable action of a finitely presented group on an ℝ-tree.

Consider an ℝ-tree T with a minimal small stable isometric action of a finitely presented group Γ. The goal of the paper is to bound the number of orbits of branch points of T and the number of orbits of directions from a branch point of T by a constant γ′(Γ) depending only on Γ. See section 1 for definitions.

Let’s be more precise. The complexity C(x) of a point x ∈ T is the number of orbits of directions at x under Stab x.

Definition. A point x ∈ T is reduced if C(x) ≠ 2, or C(x) = 2 but no direction at x is fixed by Stab x. We denote by Red(T) the set of reduced vertices of T.

Definition. A point x ∈ T is strongly reduced if one of the following holds:

1. C(x) ≥ 3

2. C(x) = 2 and no direction at x is fixed by Stab x

3. C(x) = 1 and Stab x contains a finitely generated subgroup which fixes no direction at x.

We denote by Red′(T) the set of strongly reduced vertices of T.

Remark. If (T, Γ) is minimal, and if a point x has only one orbit of directions in T, then the whole Stab (x) cannot fix any direction: otherwise there would be exactly one direction at x and T \ Γ.x would be an invariant subtree. Thus if x is reduced but not strongly reduced if and only if C(x) = 1, Stab x fixes no direction at x but every finitely generated subgroup of Stab x does.

In small actions of hyperbolic groups, every reduced point is strongly reduced (see section 1.1).

Of course, two points in the same Γ-orbit have the same complexity and are reduced (resp. strongly reduced) simultaneously. Thus, we can talk of the complexity of an orbit of reduced points x̄ = Γ.x.

1MSC 20E08

Definition. The reduced complexity C(T) of T is the sum on each orbit of reduced vertices Γ.x of the number of orbits of directions at x:

C(T) = ∑_{x̄∈Red(T)/Γ} C(x̄).

Similarly, we define the strongly reduced complexity C′(T) by

C′(T) = ∑_{x̄∈Red′(T)/Γ} C(x̄).

Clearly, bounding the (strongly) reduced complexity bounds the number of orbits of directions from a branch point and the number of orbits of (strongly) reduced branch points.

Several definitions of smallness appear in the literature. Say that a group H is algebraically small (or simply small) if H does not contain any non-abelian free group of rank 2. Say that a group H acts hyperbolically on a simplicial tree T if H contains two hyperbolic elements α, β whose axes intersect in a compact set. Say that a group H is treely small if H cannot act hyperbolically on any simplicial tree. Algebraically small implies treely small but the converse is not true: SL_{3}ℤ has property FA (i. e. fixes a point when acting on any simplicial tree) but contains a free group of rank 2. In particular, the class of treely small groups is not closed under taking subgroups. For a subgroup H of a group Γ, this condition can be relaxed in the following way: H is treely small in Γ if for any action of Γ on a simplicial tree, H cannot act hyperbolically. Of course, treely small implies treely small in any group. But if H_{1} ⊂ H_{2} ⊂ Γ, and if H_{2} is treely small in Γ, then so is H_{1}. Moreover, the class of treely small subgroups of Γ is stable under treely small extension: if H ⊲ Ĥ ⊂ Γ with H treely small in Γ, and Ĥ/H treely small, then Ĥ is treely small in Γ.

Theorem, Version 1. Let Γ be a finitely presented group. Then there exists an integer γ′(Γ) with the following property.

Consider a minimal stable action of Γ on an ℝ-tree T whose arc stabilizers are treely small in Γ. Then the strongly reduced complexity satisfies C′(T) < γ′(Γ).

Our theorem is based on accessibility of finitely presented groups. Consider Γ a finitely presented group. Consider a class C of subgroups of Γ which is stable under taking subgroups and under cyclic extension (i. e. if S ∈ C and if Ŝ ⊂ Γ contains S as a normal subgroup with Ŝ/S cyclic, then Ŝ ∈ C). Say that Γ satisfies the accessibility condition with respect to C if there exists a constant γ(Γ, C) such that every minimal action of Γ on a simplicial tree T with edge stabilizers in C has a reduced complexity C(T) bounded by γ(Γ, C).

In [BF91a], Bestvina and Feighn proved accessibility with respect to small subgroups for finitely presented groups.

Accessibility Theorem ([BF91a]). Let Γ be a finitely presented group and C the class of subgroups which are treely small in Γ. Then Γ satisfies the accessibility condition with respect to C.

Historically, the first accessibility result is Grushko Theorem [Gru40] claiming that n(A ∗ B) = n(A) + n(B) where n(Γ) is the minimum cardinal of a generating set of Γ. It follows that every minimal action of a finitely generated group Γ on a simplicial tree with trivial edge stabilizer has complexity at most 6n(Γ) − 6.

Then Dunwoody proved that a finitely presented group satisfies the accessibility condition with respect to the class of its finite groups [Dun85]. Linnell proved that the accessibility condition holds for finitely generated groups with respect to the class of its finite subgroups of order bounded by some positive number M (he actually proves something more precise, see [Lin83]).

On the other hand, there exists finitely generated groups which do not satisfy the accessibility condition with respect to the class of their small subgroups ([BF91b]) and even to the class of their finite subgroups ([Dun93]).

Concerning group actions on ℝ-trees, Jiang proved that in a free action of a finitely generated group on an ℝ-tree, there are finitely many orbits of branch points ([Jia93]). Levitt and Paulin proved that any geometric action of a finitely presented group with trivial arc stabilizer has only finitely many orbits of branch points (but this number is not bounded only in terms of Γ) [LP97]. And Gaboriau and Levitt prove that the complexity of very small action of the free group F_{n} on ℝ-trees is bounded by 6n−6 [GL95].

Our theorem bounds the complexity of group actions on ℝ-trees under an accessibility hypothesis.

Theorem, Version 2. Let Γ be a finitely presented group, and C a class of subgroups of Γ stable under taking subgroups and under cyclic extension. Assume that Γ satisfies the accessibility condition with respect to C with constant γ = γ(Γ, C).

Then there exists a constant γ′ = γ′(Γ, C) such that for every minimal stable action of Γ on an ℝ-tree T with arc stabilizers in C, the strongly reduced complexity satisfies C′(T) < γ′.

We still can be more precise. Let H = {H_{1}, …, H_{p}} be a finite collection of finitely generated subgroups of Γ. We say that Γ is finitely presented rel H if H_{i} has a presentation ⟨S_{i}|R_{i}⟩ with #S_{i} < ∞, and Γ has a presentation of the form ⟨S ⨿ S_{1} ⨿ … ⨿ S_{p}|R ⨿ R_{1} ⨿ … ⨿ R_{p}⟩ with #S < ∞ and #R < ∞. We also say that (Γ, H) is a finitely presented pair. An action of the pair (Γ, H) on an ℝ-tree T is an action of Γ in which every H_{i} ∈ H fixes a point of T. We denote by (T, Γ, H) an action of the pair (Γ, H) on T. Say that C < Γ is treely small in (Γ, H) if it cannot act hyperbolically in any action of the pair (Γ, H). Say that the pair (Γ, H) satisfies the accessibility condition with respect to C if there exists a constant γ(Γ, H, C) such that every minimal action of the pair (Γ, H) on a simplicial tree T with edge stabilizers in C has a reduced complexity bounded by γ(Γ, H, C).

It is worth noticing that the result of [BF91a] extends easily to the following result (see section 1.2):

Relative accessibility result ([BF91a]). Let (Γ, H) be a finitely presented pair. Let C be the class of treely small subgroups in (Γ, H).

Then (Γ, H) satisfies the accessibility condition with respect to C.

Theorem, Version 3. Let (Γ, H) be a finitely presented pair. Let C be a class of subgroups of Γ stable under taking subgroups and under cyclic extension. Assume that (Γ, H) satisfies the accessibility condition with respect to C with constant γ = γ(Γ, H, C).

Then there exists a constant γ′(Γ, H, C) = 15γ + γ2^{γ+dim H_{1}(Γ;ℤ/2)−1} such that for every minimal stable action of the pair (Γ, H) on an ℝ-tree T with arc stabilizers in C, the strongly reduced complexity satisfies C′(T) < γ′(Γ, H, C)

A group is slender if all its subgroups are finitely generated. Equivalently, a group is slender if any action of any of its subgroups on a simplicial tree fixes a point or a preserves line ([DS99]). Note that an action with finitely generated stabilizers of directions is stable.

Corollary. Let Γ be a finitely presented group. There exists a constant γ′(Γ) such that for every minimal action of Γ on an ℝ-tree T with slender direction stabilizer, the strongly reduced complexity satisfies C′(T) < γ′(Γ).

For a small action of a hyperbolic group, we have C(T) = C′(T).

Corollary. Let Γ be a hyperbolic group. There exists a constant γ′(Γ) such that for every minimal small action of Γ on an ℝ-tree T the reduced complexity satisfies C(T) < γ′(Γ).

This corollary is used in [LL] to prove that for any outer automorphism Φ of a non-elementary hyperbolic group, there are infinitely many classes of isogredience classes representing Φ.

The paper is organized as follows. Definitions are recalled in section 1. Section 2 deals with the complexity of graphs of actions. Section 3 shows how to reduce to the case of an almost geometric action which splits as a graph of actions with vertex actions dual to measured foliations on 2-complexes. Section 4 shows how to read the complexity of a foliated 2-complex. Sections 5, 6 and 7 bound the complexity of vertex actions of the three possible type: exotic, surface, and homogeneous. Arguments are put together in section 8 to conclude.

Acknowledgement. This paper has been inspired by a manuscript by Bestvina and Feighn ([BFa]). They contribution to the paper is thus very important. I would like to warmly thank them for that.

1 Definitions and preliminaries

In all the sequel, Γ will be a finitely generated group, H a finite collection of finitely generated subgroups of Γ so that Γ is finitely presented rel H. C will denote a class of subgroups of Γ satisfying the accessibility condition rel H, stable by taking subgroups and by cyclic extension.

1.1 ℝ-trees, directions, branch points

An ℝ-tree T is an arcwise connected metric space in which any topological arc is isometric to an interval of ℝ. Simplicial trees endowed with a path metric provide examples of ℝtrees. Let (T, Γ) be a group action on an ℝ-tree (this notation is useful when several groups act on the same tree). We will only consider actions by isometries. This action is minimal if every non-empty Γ-invariant subtree of T is T itself (we allow the trivial action of Γ on a point as a minimal action). A morphism of ℝ-trees is a continuous map f : T → T′ such that every compact interval of T can be subdivided into finitely many sub-arcs on which f is isometric. Given (T_{1}, Γ_{1}), (T_{2}, Γ_{2}) and a morphism φ : Γ_{1} → Γ_{2}, a morphisms of ℝ-trees f : T_{1} → T_{2} is φ-equivariant (or simply equivariant) if f(γ.x) = φ(γ).f(x). All the morphisms we consider are equivariant.

A direction at x ∈ T is a connected component of T \ {x}. Thus, two points y_{1}, y_{2} are in the same direction from x if [x, y_{1}] ∩ [x, y_{2}] contains more than one point. A vertex of T is either • a branch point, i. e. a point x ∈ T from which there are at least 3 directions,

• a flip point, i. e. a point x with exactly two directions and such that there exists γ ∈ Stab x exchanging the two directions

• or a terminal point, i. e. a point x ∈ T from which there is just one direction.

Note that in a minimal action of a finitely generated group, there are no terminal points, and there are at most countably many vertices. A flip point is always reduced. A reduced point is always a vertex.

Lemma 1.1. If arc stabilizers are finitely generated and satisfy the ascending chain condition, then any reduced point is strongly reduced.

Remark. Small actions of hyperbolic groups satisfy the hypotheses of this lemma.

Proof. Because arc stabilizers satisfy the ascending chain condition, every direction stabilizer fix a non-degenerate arc.

Take x a point with C(x) = 1. Let H_{0} be the stabilizer of a direction η_{0} at x. By minimality, there exists γ ∈ Stab x such that γ.η ≠ η. If x is not strongly reduced, then the finitely generated group < γ, H_{0} >⊂ Stab x fixes a direction η_{1} at x. The group H_{1} = Stab η_{1} strictly contains H_{0}. By induction, we construct an infinite ascending chain of arc stabilizers giving a contradiction.

1.2 Reduced complexity

The original statement of [BF91a] is the following:

Theorem ([BF91a]). Let Γ be a finitely presented group. Then there exists a constant γ(Γ) such that the following holds.

Consider a minimal action of Γ on a simplicial tree with treely small edge stabilizers such that every vertex is reduced. Then T has at most γ(G) orbits of vertices.

The following remark is due to G. Levitt. We assume there is no vertex (in the simplicial sense) in T with exactly 2 incident edges which is not a flip point. Consider a non-reduced vertex v. Then there is exactly one edge e incident on v with same stabilizer as v. We say that this edge is non-reduced. Consider the tree T′ obtained by collapsing every non-reduced edge in T. Then T′ has same reduced complexity as T and every vertex of T′ is reduced. This allows to deduce the accessibility theorem given in the introduction.

Let’s give a check of the proof the relative version of the accessibility result. A Dunwoody resolution of a minimal simplicial action (T, Γ) is simplicial action (S, Γ) with an equivariant map f : S_{0} → T which embeds every edge and which is simplicial with respect to some equivariant subdivision of S. The main result of [BF91a] may be restated this way:

Assume that (T, Γ) is a minimal simplicial action of a finitely generated group. Assume that T has a Dunwoody resolution (S, Γ) with at most n orbits of vertices. Assume that edge stabilizers of T don’t act hyperbolically on S.

Then the reduced complexity satisfies C(T) ≤ 188n + 468β_{1}(Γ) + 12 dim H^{1}(Γ; ℤ/2) − 280.

For a finitely presented group, the existence of a bound δ(Γ) such that every minimal simplicial action (T, Γ) has a Dunwoody resolution with at most δ(Γ) orbits of vertices is proved in [Dun85]. This immediately extends to actions of a finitely presented pair: Γ is finitely presented rel H = {H_{1}, …, H_{p}} if it acts freely on a simply connected 2-complex X containing simply connected sub-complexes X_{i} whose stabilizer is H_{i}, whose orbits are disjoint, and such that [X \ (∪_{i}X_{i}^{(2)})]/Γ is finite. We first construct an equivariant map f : X → T: send X_{i} to a point fixed by H_{i}, and extend it equivariantly to X^{(0)}. Extend f linearly on edges, and extend f to X^{(2)} so that for each closed 2-simplex σ, either f is constant on σ or its level sets are the leaves of a foliation which is either regular or has a single 3-pronged singularity in its interior.

We consider the induced measured foliation on X (where simplices on which f is constant are understood as being contained in a leaf). X being simply connected, the space of leaves of X is a simplicial tree S and f : X → T induces a Dunwoody resolution. The preimage of a vertex of S contains at least a vertex of X or a singularity in a 2-dimensional simplex. Thus, the number of vertices of S is bounded by # (X^{(0)}/Γ) + # [(X^{(2)} \ (⋃_{i} X_{i}^{(2)})) /Γ]. Therefore, the accessibility result for relatively finitely presented groups holds.

1.3 Stable actions.

An interval (or a subtree) I ⊂ T is non-degenerate if it contains more than a point. A non-degenerate interval I is stable if for every non-degenerate subinterval J ⊂ I, one has equality between pointwise stabilizers Stab I = Stab J. The action (T, Γ) is stable if every non-degenerate interval in T contains a stable interval. For example, if the set of arc stabilizers satisfy the ascending chain condition, then the action is stable. In this case the stabilizer of a direction fixes a non-degenerate arc.

2 Complexity of a graph of actions

2.1 Graph of actions

Definition. A graph of actions on ℝ-trees G is a graph of groups with vertex groups Γ_{v}, edge groups Γ_{e} and edge morphisms i_{e} : Γ_{e} → Γ_{t(e)} together with the following data:

• for each vertex v, an action (T_{v}, Γ_{v}) of the corresponding vertex group

• for each oriented edge e incident to v = t(e), an attaching point p_{e} ∈ T_{v} fixed by i_{e}(Γ_{e})

• a (maybe 0) length for each non-oriented edge of Q.

We define the fundamental group of G to be the fundamental group of the underlying graph of groups. To a graph of actions G naturally corresponds an action (T_{G}, π_{1}(G)): it is obtained from the universal cover G̃ of the graph of groups underlying G by replacing a vertex ṽ of G̃ by a copy of the corresponding vertex ℝ-tree T_{v}, by gluing equivariantly edges incident to a vertex v on T_{v} according to the attaching points, and by collapsing the 0-length edges (see for instance [Lev94, Gui98]).

We say that an action (T, Γ) splits as a graph of actions if (T, Γ) is equivariantly isometric to the action (T_{G}, π_{1}(G)) corresponding to a graph of actions G. If T = T_{G} is a splitting of T as a graph of actions, then consider the equivariant family S of nondegenerate subtrees of T corresponding either to a non-degenerate vertex tree of G or to an edge with positive length. Then S covers T and for S, S′ ∈ S, S ∩ S′ contains at most one point. Moreover, every arc in T is covered by finitely many elements of S. There is a reciprocal to this observation:

Lemma 2.1. Let (T, Γ) be an action of a group on an ℝ-tree. Assume that T is covered by a family of closed non-degenerate subtrees S invariant under the action of Γ such that

• for S ≠ S′ ∈ S, S ∩ S′ contains at most one point

• every arc in T is covered by finitely many elements of S.

Then T has a natural decomposition into a graph of actions.

Remark. The graph of actions obtained satisfies that

• the vertex action of an endpoint of an edge of positive length is a point

• edges of length 0 have exactly one endpoint whose vertex action is a point

• If e_{1}, e_{2} are two different length-0 edges incident on the same vertex v such that T_{v} is non-degenerate then p_{e_{1}} and p_{e_{2}} are not in the same orbit in (T_{v}, Γ_{v})

We say that a graph of actions obtained by this lemma is a graph of actions in standard form.

Proof. We say that an arc in an ℝ-tree a simplicial arc if it doesn’t meet any branch point of T except maybe at its endpoints. Let E_{1} be the subset of S consisting of simplicial arcs which are not flipped by any element of Γ. Let V_{1} = S \ E_{T}. Let V_{0} = {x ∈ T | ∃S ≠ S′ ∈ S s.t. x ∈ S ∩ S′} and let E_{0} = {(x, S) ∈ V_{0} × V_{1}| x ∈ S}. If (T, Γ) is not minimal, we may need to add to V_{0} some endpoints of simplicial arcs in E_{1} if they are have valence 1 in T. V = V_{1} ∪ V_{0} and E = E_{1} ∪ E_{0} are the sets of vertices and non-oriented edges of a graph G̃: the endpoints of e = (x, S) ∈ E_{0} are x ∈ V_{0} and S ∈ V_{1}, and the endpoints of S = [a, b] ∈ E_{1} are the points a, b ∈ V_{0}. We assign the length 0 to any edge in E_{0}/Γ, and we assign the positive length of the corresponding simplicial arc in S to any edge in E_{1}/Γ. Clearly, G̃ is a tree endowed with an isometric action of Γ. Therefore, we naturally get a graph of actions whose graph of groups is G̃/Γ, whose vertex actions are conjugacy classes of the actions (S, Stab (S)) for S ∈ V.

2.2 The complexity of a graph of actions

Lemma 2.2. Consider a graph of actions on ℝ-trees G in standard form, and assume that every vertex action is minimal.

Let A be the set of oriented (length-0) edges e ∈ E(G) whose attaching point p_{e} is not reduced in the corresponding vertex tree.

Let B_{1} be the set of vertices v ∈ V_{0}(G) of valence 2 in G such that, both incident edges have positive length and at least one of the edge morphisms is onto.

Let B_{2} be the set of vertices v ∈ V_{0}(G) of valence 2 in G such that, some edge e incident on v has positive length and i_{e} is onto, and the other edge e′ incident on v has length 0, and p_{}_{ē′} has exactly one orbit of directions in T_{t(ē′)}.

Let a = #A, b_{1} = #B_{1} and b_{2} = #B_{2}. Then

C(T_{G}) = ∑_{v} C(T_{v}) + 2#{non-oriented edges of positive length} + 2a − 2(b_{1} + b_{2}).

Equivalently, if G_{0} denotes the graph of groups obtained from G by collapsing length-0 edges and G̃_{0} its Bass-Serre universal cover,

C(T_{G}) = C(G̃_{0}) + ∑_{v} C(T_{v}) + 2a − 2b_{2}.

Remark. If vertex actions are not minimal then an additional term has to be included because some other vertices in V_{0}(G) may become non-reduced in T_{G}.

Proof. Consider (T_{G}, Γ) the action corresponding to the graph of actions G. Let S be the collection of non-degenerate subtrees of T_{G} corresponding the decomposition of T_{G} into a graph of actions. For x ∈ T_{G}, let S_{x} be the set of S ∈ S containing x. For S ∈ S_{x}, let Γ_{S} denote the setwise stabilizer of S, and let C_{S}(x) be the number of orbits of directions of x in (S, Γ_{S}). Since any arc [x, x′] is covered by finitely many elements of S, any direction at x ∈ T_{G} corresponds to a direction at x in some S ∈ S_{x}. Thus,

C(x) = ∑_{S∈S_{x}/Stab (x)} C_{S}(x).

Hence,

C(T_{G}) = ∑_{x∈Red(T_{G})/Γ} C(x) = ∑_{x∈Red(T_{G})/Γ} ∑_{S∈S_{x}/Stab (x)} C_{S}(x).

Therefore, the complexities C(S) for S ∈ S/Γ add up to the complexity of T (G) except that a point x which is reduced in T_{G} might not be reduced in T_{S} or vice-versa. This translates into the following equality:

C(T_{G}) = ∑_{S∈S/Γ} C(S) + ∑_{{(x,S)∈E_{0}(T_{G})/Γ|x∉Red(S)}} C_{S}(x) − ∑_{{x∈V_{0}(T)/Γ|x∉Red(T_{G})}} C(x)

= ∑_{S∈S/Γ} C(S) + 2#{(x, S) ∈ E_{0}(T_{G})/Γ|x ∉ Red(S)} − 2#{x ∈ V_{0}(T)/Γ|x ∉ Red(T_{G})}

Since a = #{(x, S) ∈ E_{0}(T_{G})/Γ|x ∉ Red(S)}, we have to prove that x ∈ V_{0}(T) is not reduced in T_{G} if and only if the corresponding vertex v ∈ G lies in B_{1} ∪ B_{2}.

A point x ∈ V_{0}(T_{G}) is not reduced in T_{G} if and only if C(x) = 2 and Stab (x) fixes a direction η_{0} in T_{G}. Thus Stab (x) preserves the tree S_{0} ∈ S_{x} intersecting this direction. Since x ∈ V_{0}(T_{G}), x lies in some S_{1} ∈ S_{x} \ {S_{0}}. Since C(x) = 2, C_{S_{1}}(x) = C_{S_{0}}(x) = 1, and S_{x} = {S_{0}, Stab (x).S_{1}}. In particular, v is a valence 2 vertex in G. Moreover, η_{0} is the only direction at x in S_{0}, so S_{0} cannot be minimal, and by hypothesis, S_{0} is a simplicial arc. It is now clear that if S_{1} is a simplicial arc then v ∈ B_{1}, and v ∈ B_{2} otherwise.

We will use the previous Lemma in the following form:

Corollary 2.3. Let G and G′ be 2 graphs of actions on ℝ-trees having the same underlying metric graph of groups G. Assume that every vertex action of G and G′ is minimal and that any arc stabilizer in (T_{v}, Γ_{v}) fixes an arc in (T′_{v}, Γ_{v}).

Then we can move attaching points of G′ so that

C(T_{G}) ≤ C(T′_{G}) + ∑_{v∈V (G)} [C(T_{v}) − C(T′_{v})] + 2C(G̃_{0})

where G_{0} is the graph of groups obtained from G by collapsing edges of length 0, and G̃_{0} denotes its universal cover.

Proof. Lemma 2.2 implies that

C(T_{G}) = C(T_{G′}) + ∑_{v∈V (G)} [C(T_{v}) − C(T′_{v})] + 2(a − a′) − 2(b_{2} − b′_{2}).

We can manage so that a′ = a in the following way: consider an edge e ∈ E(G) having an attaching point p_{e} which is not reduced in T_{v} where v = t(e). The stabilizer of p_{e} in (T_{v}, Γ_{v}) fixes an arc in T_{v}, so it also fixes an arc I′ in T′_{v}. Thus, we can change the attaching point p′_{e} of e in T′_{v} to any point in I′, so we can choose an attaching point which is not one of the countably many vertices. This way we get a = a′.

Since B_{1} = B′_{1}, we just need to prove that b′_{2} ≤ C(G̃_{0}). Consider G̃ the universal cover of G. We first prove that if v is any vertex in G̃ such that T_{v} is non-degenerate, then its image v̄ in G̃_{0} is reduced. Let x ∈ T_{v} such that (x, T_{v}) ∈ E_{0}(T_{G}). By minimality of vertex actions, there is an element of Γ_{v} which does not fix x. This means that Γ_{v} fixes no edge in G̃. Since the map from G̃ to G̃_{0} consists in collapsing length-0 edges, we get that Γ_{S} fixes no edge in G_{0}. Therefore, the image of v in G̃_{0} is reduced.

Let v ∈ B_{2}, let e′ be the length-0 edge incident on v and let v′ = t(ē′). T_{v′} is minimal and non-degenerate, hence the image of v′ is reduced in G_{0}. Consider the map from B_{2} to E(G_{0}) sending a vertex v ∈ B_{2} to the image in G_{0} of the edge positive length incident on v in G. Then its image is contained in the set of oriented edges whose endpoint is reduced. Since this map is one to one, we get that b′_{2} ≤ C(G̃_{0}).

3 Strong approximations, finitely presented pairs and almost geometric actions.

The goal of this section is to reduce the proof of the Theorem to actions dual to pure minimal systems of isometries.

3.1 Strong approximations are not less complex.

In this section we prove that if an action (T, Γ, H) is a strong limit of a sequence of actions whose reduced complexities are bounded by M, then the strongly reduced complexity of (T, Γ, H) is also bounded by M.

Definition ([LP97]). Let {Γ_{i}}_{i∈ℕ} be a sequence of finitely generated groups with epimorphisms φ_{ij} : Γ_{i} ↠ Γ_{j} for i < j so that φ_{jk} ∘ φ_{ij} = φ_{ik} for i < j < k. Let Γ be the direct limit lim_{→} Γ_{i} and φ_{i} : Γ_{i} ↠ Γ the natural morphisms.

A sequence of minimal actions on ℝ-trees (T_{i}, Γ_{i}) converges strongly to (T, Γ) if

• for every i < j, there are equivariant morphisms of ℝ-trees f_{ij} : T_{i} → T_{j} and f_{i} : T_{i} → T

• for every i < j < k, f_{jk} ∘ f_{ij} = f_{ik} and f_{j} ∘ f_{ij} = f_{i}

• for every finite subtree^{2} K in some T_{i}, there exists j > i such that f_{j} is an isometry in restriction to f_{ij}(K).

2a finite subtree is the convex hull of finitely many points

Remark. When Γ is finitely presented, the morphisms φ_{ij} and φ_{i} are isomorphisms for large enough i, so Γ_{i} = Γ. We will see that an action of a finitely presented pair (Γ, H) is a strong limit of (almost-)geometric actions of the pair (Γ, H) i. e. Γ_{i} = Γ, φ_{ij} = Id, and elements of H are elliptic (see lemma 3.3, prop. 3.4).

Proposition 3.1. Let (T, Γ, H) be an action of a finitely presented pair whose strongly reduced complexity satisfies C′(T) ≥ N for some finite number N. Assume that (T_{i}, Γ, H) converges strongly to (T, Γ, H). Then for i sufficiently large, the reduced complexity of T_{i} satisfies C(T_{i}) ≥ N.

Remark. We don’t assume a priori that the complexity of T is finite but this is a consequence of Theorem 1 when T is a small stable action.

Proof. Consider a sequence (T_{i}, Γ, H) converging strongly to (T, Γ, H). Denote by f_{i} : T_{i} → T the corresponding morphisms of ℝ-trees.

It is a standard fact that the definition of strong convergence implies that for any finite tree K ⊂ T and any finite subset F ⊂ Γ, we can isometrically lift K and the action of F to some K_{i} ⊂ T_{i} in the following sense: fi|K_{i} is an isometry between K and K_{i}, and for every γ ∈ F and every a, b ∈ K_{i}, γ.a = b if and only if γ.f_{i}(a) = f_{i}(b).

Indeed, take a finite set {x_{1}, …, x_{p}} ⊂ T whose convex hull is K. Choose a preimage y_{k} of each x_{k} in T_{1}. Let K_{1} bet the convex hull of {γ.y_{k} | γ ∈ F, k = 1, …, p}. Then for i large enough, K_{i} = f_{1i}(K_{1}) embeds into T and provides the desired lift.

Claim 3.2. Consider a strongly reduced vertex x ∈ T, and a finite number n ≤ C(x). Then for large enough i, there is a preimage x′ of x in T_{i} such that x′ is a reduced vertex of T_{i} and C(x′) ≥ n.

The claim implies the proposition since if x_{1}, …, x_{p} ∈ T denote some strongly reduced vertices in distinct orbits such that C(x_{1}) + ⋯ + C(x_{p}) ≥ N then we find some reduced points x′_{1}, …, x′_{p} ∈ T_{i} for some i such that C(x′_{1}) + ⋯ + C(x′_{p}) ≥ N. Moreover, these vertices are in distinct orbits since f_{i}(x′_{k}) = x_{k} and f_{i} is equivariant.

So let’s turn to the proof of the claim. Consider a strongly reduced vertex x ∈ T and n ≤ C(x) (take n = C(x) if C(x) < ∞ and n ≥ 3 if C(x) = ∞). Take a finite subtree K ⊂ T containing {x} and at least n directions at x which are not in the same orbit under Stab (x). Take i big enough so that K lifts to some K_{i} ⊂ T_{i} such that f_{i} restricts to an isometry on K_{i} and let x′ = (f_{i}|K_{i})^{−1}(x). Thus C(x′) ≥ n. So we just have to check that x′ is a reduced vertex for large enough i.

Let’s first check that x′ is a vertex in T_{i}. If x is a branch point of T, then n ≥ 3 and x′ is branch point in K_{i} and therefore in T_{i}. If x is a flip point with g flipping the two directions at x, we take K containing the two directions at x and i large enough so that g flips the two directions at x′ in K_{i}. If x′ is not a branch point in T_{i}, then it is a flip point and hence a vertex.

Let’s prove that x′ is reduced. If x has at least 3 orbits of directions, then so does x′. If x has exactly two orbits of directions (say Stab (x).η_{1} and Stab (x).η_{2} with η_{1}, η_{2} ⊂ K), then there exists g_{1}, g_{2} ∈ Stab (x) which don’t fix η_{1} and η_{2} respectively. Take i big enough so that g_{1} and g_{2} fix x′. Since f_{i} is equivariant, g_{1} and g_{2} respectively don’t fix the preimage under f_{i}|K_{i} of η_{1}, η_{2} so x′ is reduced.

Finally, assume that x has only one orbit of directions. Since x is strongly reduced, there exists a finitely generated group H which fixes no direction at x. We assume that i large enough so that H fixes x′. If x′ is not reduced, then Stab x′ fixes a direction at x′, and so does H. Hence H fixes a direction at x, a contradiction.

The conclusion is that x′ is reduced for large enough i, and C(x′) ≥ N.

3.2 Strong approximation by an almost geometric action.

We restate here the fact any action has a strong approximation by a geometric action and that a geometric action splits into pure components (see [Gui98, prop. 4.1] or [BF95], [GLP94]).

Foliated 2-complexes, geometric actions, systems of isometries.

Consider a foliated 2-complex Σ (see [LP97]). Roughly speaking, a foliated 2-complex is a 2-dimensional simplicial complex where 2-simplices are endowed with a regular measured foliation, and 1-simplices are assigned a measure the set of which is invariant under holonomy.

Edges may be transverse to the foliation or contained in a leaf. Given a finitely generated group Γ and a morphism ρ : π_{1}Σ → Γ, consider the corresponding Galois covering Σ̄ of Σ. The transverse measure of the lifted foliation on Σ̄ induces a pseudometric on Σ̄. The space TΣ̄ obtained by making this pseudo-metric Hausdorff is called the leaf space made Hausdorff of Σ̄. We say that leaf space of Σ̄ is Hausdorff to mean that two different leaves are at a non-zero distance for this pseudo-metric. When ker ρ is normally generated by free homotopy classes of loops contained in leaves of Σ, TΣ̄ is an ℝ-tree ([Lev93a, LP97]).

Definition. When every component of the preimage D̄ of D in Σ̄ isometrically embeds into TΣ̄, we say that Σ̄ is tame.

Definition. An action of a finitely generated group (T, Γ) is geometric if it is isomorphic to some (TΣ̄, Γ) where Σ̄ is tame (the leaf space of Σ̄ is not required to be Hausdorff) (see [LP97]). We sometimes say that (T, Γ) is dual to Σ.

Many foliated 2-complexes are obtained as the suspension of a system of isometries. A system of isometries X on a metric graph D is a finite set of partially defined isometries between non-empty compact connected subsets of D. A partial isometry φ ∈ X is called a generator. A generator is a singleton if its domain contains exactly one point. When D is a multi-interval (that is a finite union of compact intervals), there is a corresponding open system of isometries X^{∘} which is the restriction of the generators of X to the interior of their domain. We say that X is pure if X contains no singleton and if the X^{∘}-orbit of every point in D^{∘} is dense in D.

The suspension Σ of X is the foliated 2-complex obtained by gluing on D, for each generator φ ∈ X, a band (dom φ) × [0, 1] where (x, 0) and (x, 1) are glued with x and φ(x) respectively. Each band is foliated by {∗} × [0, 1], and we consider the transverse measure which gives to every arc of D a measure equal to its length.

If an action (T, Γ) is geometric, then it is dual to the suspension of a system of isometries induced by the generators of Γ on a finite subtree K ⊂ T ([LP97]). Moreover, if Γ is finitely presented and if H′_{1}, …, H′_{p} < Γ are finitely generated subgroups which are elliptic in T, then it is dual to a foliated 2-complex Σ where π_{1}(Σ̄) is normally generated by curves contained in leaves, and each H′_{i} preserves a leaf in Σ̄. Such a presentation of the action is said to be in standard form.

Strong approximations of actions of a finitely presented pair

We now prove that actions of a finitely presented pair (Γ, H) have strong approximations by geometric actions of the pair (Γ, H) (i. e. Γ_{i} = Γ, φ_{ij} = Id, and elements of H are elliptic in T_{i}).

Lemma 3.3. Let (T, Γ, H) be a minimal action of a finitely presented pair. Then (T, Γ, H) is a strong limit of geometric actions of the pair (Γ, H).

Proof. Start with a sequence (T_{i}, Γ_{i}) of geometric actions of finitely presented groups converging strongly to (T, Γ). Since H is made of finitely many finitely generated groups, by lifting the action of its generators to T_{i} we can assume that for i big enough, there are finitely generated groups H′_{1}, …, H′_{p} < Γ_{i} such that φ_{i}(H′_{k}) = H_{k} and H′_{k} preserves a leaf in Σ̄_{i}.

Consider Σ_{i} a foliated 2-complex and ρ_{i} : π_{1}Σ_{i} → Γ_{i}, so that T_{i} is the leaf space made Hausdorff of the corresponding Galois covering Σ̄_{i}. We take this presentation of T_{i} in standard form so that we can assume that π_{1}Σ̄_{i} is normally generated by curves contained in leaves and H′_{k} fix a leaf in Σ̄_{i}.

Since (Γ, H) is finitely presented, ker φ_{i} is normally generated by subgroups of H′_{1}, …, H′_{p} for large enough i. Hence, Σ̄_{i}/ ker φ_{i} is normally generated by curves contained in leaves. So, its leaf space made Hausdorff T′_{i} is an ℝ-tree, endowed with an action of the pair (Γ, H). Since the map f_{i} : T_{i} → T factors through T′_{i}, the sequence of geometric actions (T′_{i}, Γ, H) converges strongly to T_{i}.

Almost geometric actions

Definition. An action of a finitely generated group on an ℝ-tree is almost geometric if it splits as a finite graph of actions on ℝ-trees G such that for every vertex action (T_{v}, Γ_{v}), there is a normal subgroup N_{v} ⊲ Γ_{v} contained in the kernel of (T_{v}, Γ_{v}) and (T_{v}, Γ_{v}/N_{v}) is geometric. We use the notation Γ_{v}^{0} = Γ_{v}/N_{v}.

Remark. If the subgroups N_{v} are trivial, then the action is geometric (see [Gui00]).

An almost geometric action needn’t be geometric: a free action of ℤ^{n} on the real line by translations is geometric. Now consider the action of the free group F_{n} on the real line induced by a morphism F_{n} → ℤ^{n}. The obtained action of F_{n} is not geometric since it is proved in [GL95] that in a geometric action of F_{n}, the set of fixed points of any element is compact.

The following proposition is a restatement of the fact that a system of isometries splits into pure components, and that a minimal component gives an action whose arc stabilizers are contained in its kernel (under a stability hypothesis). It was proved in the setting of finitely presented group in [Gui98] but adapts adapts immediately to finitely presented pairs using 3.3 ([Gui98] Prop. 4.1, Lemma 4.2).

Proposition 3.4 (D. Gaboriau, [Gui98], prop 4.1). A stable action (T, Γ, H) of a finitely presented pair is a strong limit of almost geometric actions (T_{i}, Γ, H) having a nice decomposition in the sense of the definition below.

Definition 3.5. An almost geometric action has a nice decomposition if it decomposes into a graph of actions (T_{G}, Γ) with the following properties (using notations of the previous definition).

• Vertex groups Γ_{v} are finitely generated

• Every action (T_{v}, Γ_{v}^{0}) is minimal (it may be degenerate), it has trivial arc stabilizers, and its orbits are dense in every segment.

• If T_{v} is non-degenerate, then the following holds: there is a pure minimal system of isometries X_{v}, a Γ_{v}^{0}-cover Σ̄_{v} of its suspension Σ_{v}, such that π_{1}(Σ̄_{v}) is normally generated by curves contained in leaves, and (T_{v}, Γ_{v}^{0}) is the leaf space made Hausdorff of Σ̄_{v}.

• Σ̄_{v} is tame and the leaf space of Σ̄_{v} is Hausdorff.

In view of Lemma 2.2 and Proposition 3.1 we want to analyze actions coming from pure minimal system of isometries.

3.3 Pruning process and the 3 types of pure systems of isometries

There are 3 exclusive types of pure systems of isometries: homogeneous type, surface type, and exotic type. When the X^{∘}-orbits are locally the trace of a group of isometries of ℝ, then X is said to be homogeneous (or axial). This group of isometries is well defined by X up to conjugacy. It is called the group of periods of X. An homogeneous system of isometries may either be orientable or non-orientable according to the fact that its group of periods is orientable or not. If X is homogeneous and if TΣ̄ has trivial arc stabilizers, then TΣ̄ is a line, and Γ is isomorphic to the group of periods (see [Gui98], [BF95], [Pau97]).

For the two other types of system of isometries we need to recall the definition of the pruning process (or Rips Machine 1). A Rips move is an operation on X, D and Σ which doesn’t change TΣ̄, tameness, and the fact that the space of leaves of Σ̄ is Hausdorff. Thanks to a theorem by Gaboriau, given a non-homogeneous system of isometries, one can perform Rips moves on it so that it generators become independent: this means that no word in the X^{∘}-generators fixes a point in D (see [Gab97, Pau97, Gui98]). A system of isometries with independent generators has the following property: the total length of the domain D equals the sum of the lengths of the domains of the generators ([Lev93b, GLP94]). This means that

• either every point of D but finitely many of them lie in the domain of exactly 2 generators,

• or the set E of points of D which lie in the domain of exactly one generator is non-empty.

In the first case, Σ is a surface with boundary with a measured foliation. In the second case, E is a finite union of intervals; these intervals are open in D and their closures don’t intersect by purity of X. So we can prune E: we define a new system of isometries whose domain is D′ = D \ E, and the new set of generators consists in the restrictions of generators of X to D′. This pruning operation is a Rips move, generators remain independent, and X′ is still pure ([Gab96]). Note that the pruning operation removes all the points in D which are terminal vertices in their leaf.

Therefore, either the suspension of the new system of isometries is a surface, or the pruning operation can be iterated. If this pruning process stops, then the final 2-complex is a surface and we say that X is of surface type (or interval exchange). Otherwise, X is called exotic (thin in [BF95]).

We say that a pruning operation is an interior pruning if E doesn’t intersect ∂D. According to [BF95, Gab96], all the prunings but finitely many of them are interior prunings.

4 Reading the complexity from the foliation.

Consider X a system of isometries, Σ it suspension, ρ : π_{1}Σ ↠ Γ, and TΣ̄ the dual tree. The goal of this section is to compute the complexity of TΣ̄ in terms of Σ when Σ̄ is tame, and its leaf space is Hausdorff. This section follows [GL95] which deals with actions of a free group. Notations are those of section 3.2.

Let x be a point in TΣ̄, \overline{L} the corresponding leaf in Σ̄, and L its projection in Σ. L and \overline{L} come with their natural graph structure. Note that Stab (x) is conjugate to ρ(i_{∗}(π_{1}(L))) where i : L ↪ Σ denotes the inclusion.

Now consider the graph of directions DL of L: a vertex of DL is a direction in D from a point v ∈ L ∩ D. Here we think of a direction at v as a germ of isometric map d : [0, ε] → D with d(0) = v. We put an edge between two directions d, d′ at v, v′ for every generator φ such that φ ∘ d is defined on a non-degenerate interval and d′ = φ ∘ d. We denote by q : DL → L the natural map sending d to d(0). Similarly, there is a graph of directions \overline{DL} of \overline{L}: its vertices are directions from a point v ∈ \overline{L} in D̄, we put an edge between d and d′ for each band between d(0) and d′(0) whose holonomy sends d to d′. We denote by q̄ the natural map \overline{DL} → \overline{L}. When the leaf space of Σ̄ is Hausdorff, we have Stab (\overline{L}) = Stab (x), and DL = \overline{DL}/Stab (\overline{L}).

The following lemma computes the complexity of x ∈ TΣ̄ in terms of the graph of directions at L. This is a direct generalization of Lemma III.5 in [GL95].

Lemma 4.1. We assume that Σ̄ is tame, that the leaf space of Σ̄ is Hausdorff, and that arc stabilizers of TΣ̄ are trivial.

• The set of directions at x ∈ TΣ̄ is in one-to-one correspondence with the set of connected components of \overline{DL}.

• The set of orbits of directions at x is in one-to-one correspondence with the set of connected components of DL.

Since a complexity of a point an hence of the action can be read off Σ we will denote by C(p) the number of components of DL, the graph of directions at a point p ∈ Σ. We will also use the abuse of notation C(Σ) = C(TΣ̄).

Corollary 4.2. If L is a regular leaf, then x is not a branch point. In this case, x is a flip point if and only if L is transversally non-orientable.

Proof of the corollary. If L is regular, then \overline{DL} is a 2-covering of \overline{L}. Moreover, \overline{DL} is not connected because Σ̄ is tame. Thus DL is a the disjoint union of 2 copies of L, so x is not a branch point by the lemma. Now x is a flip point if and only if an element of Stab (x) exchanges those two copies. This means that L is transversally non-orientable.

Proof of the lemma. Since Σ̄ is tame, there is a natural map Δ from \overline{DL} to the set of directions at x in TΣ̄: just project on TΣ̄ a germ d̄: [0, ε] → D̄. Lemma 4.3 will say that Δ is onto.

If d̄, d̄′ are in the same component of \overline{DL}, then there is a segment in L joining q(d̄) to q(d̄′) whose holonomy sends d to d̄′ so that Δ(d̄) and Δ(d̄′) define the same direction in TΣ̄.

Conversely, if Δ(d̄) = Δ(d̄′) then for η small enough, ∀t ∈ [0, η], d̄(t) and d̄′(t) are in the same leaf. By segment-closed property (see [GLP95, Th.2.3]), there exists η′ < η and a word w in the generators of X, such that the projections d, d′ of d̄, d̄′ in Σ are such that w ∘ d′(t) = d(t). Thus w lifts to a band of leaves whose holonomy sends d̄ to some γ.d̄′. Therefore, for t < η′, γ.d̄′(t) and d̄′(t) are in the same leaf. This means that γ fixes an arc in TΣ̄ so γ = 1. Thus the band of leaves defined by w provides a path in DL joining d̄ to d̄′. This proves the first part of the lemma. The second part is clear since Δ is Stab (x)-equivariant.

Lemma 4.3. Any germ of isometric map d : [0, ε] → TΣ̄ can be lifted to D̄.

Remark. In this lemma, the fact that the leaf space of Σ̄ is Hausdorff is not necessary.

Proof. Consider two preimages a, b of d(0), d(ε) in D̄. Take a path α joining a to b in Σ̄ which is a concatenation of subpaths contained in a leaf or in D̄. Let t_{0} be the last instant for which α(t_{0}) maps to d(0) in TΣ̄. Then for η small enough, the restriction of α to [t_{0}, t_{0} + η] is a desired lift.

5 Complexity of exotic components.

Proposition 5.1. Consider (T, Γ, H) an almost geometric action of a pair with a nice decomposition as a graph of actions G (definition 3.5). Then, for each exotic vertex action (T_{v}, Γ_{v}^{0}), there is a simplicial action (T′_{v}, Γ_{v}^{0}) with trivial arc stabilizer, in which every point stabilizer of (T_{v}, Γ_{v}^{0}) fix a point, and satisfying

C(T_{v}) ≤ C(T′_{v}).

Furthermore, there is G′ be a graph of actions obtained from G by changing T_{v} to T′_{v} such that (T_{G′}, Γ, H) is an almost geometric action with no exotic component, its arc stabilizers are in C if those of T are, and we have

C(T) ≤ C(T_{G′}) + 2γ.

Remark. We actually construct T′_{v} as an approximation of T_{v}.

Lemma 5.2. Let Σ be the suspension of a pure exotic system of isometries, and let {L_{1}, …, L_{n}} be a finite set of leaves of Σ. Then there exists a foliated 2-complex Σ′ whose leaves are compact, a finite set {L′_{1}, …, L′_{n}} of leaves of Σ′ and a map f : Σ′ → Σ with the following properties.

(i) f is a homotopy equivalence, f(D′) ⊂ (D), f is an isometry in restriction to each component of D′, and f sends any leaf of Σ′ to a subset of a leaf of Σ.

(ii) ∀i = 1, …, n, f(L′_{i}) ⊂ L_{i} and f_{∗} : π_{1}(L′_{i}) → π_{1}(L_{i}) is onto

(iii) C(Σ) ≤ C(Σ′)

Proof of the Proposition using the Lemma. Consider a nice decomposition (T_{G}, Γ) of T. For each vertex v corresponding to a pure exotic system of isometries Σ_{v}, let Σ′_{v} be given by the lemma, where {L_{1}, …, L_{n}} is the set of leaves corresponding to attaching points of edges incident on v in the graph of actions. Let T_{v} = TΣ̄_{v} and T′_{v} = TΣ̄′_{v}. f induces an equivariant morphism of ℝ-trees between (T_{v}, Γ_{v}^{0}) and (T′_{v}, Γ_{v}^{0}) showing that edge stabilizers of (T′_{v}, Γ_{v}^{0}) are trivial. Property (ii) says that stabilizers of attaching points of T_{v} fix a point in T′_{v}.

Let G′ be a graph of actions obtained by replacing T_{v} by T′_{v} for each exotic vertex v. In view of corollary 2.3, we may move attaching points to get C(T) ≤ C(T_{G′}) + 2γ.

Proof of the Lemma. Following [Gui98], because generators are independent, any loop in a leaf is either nullhomotopic or contains a singular edge, i. e. one of the two sides of a band which is contained in a leaf. Thus, any non simply-connected leaf L is one of the finitely many singular leaves, and it contains a finite graph K(L) whose fundamental group generates π_{1}(l) (if L is simply connected, we set K(L) to one point).

We will narrow a band by a small amount δ as in [Gui98, BFb]: given an edge e = {b}×[0, 1] in the boundary of a band [a, b]×[0, 1], narrowing this band at e means replacing it by [a, b − δ] × [0, 1]. The obtained foliated 2-complex is denoted by Σ_{δ} and L_{e} denotes the Σ-leaf containing e. The condition (i) for inclusion Σ_{δ} ⊂ Σ requires that the map induced by inclusion L_{e} \ e ↪ L_{e} on fundamental groups is onto. During iteration of the pruning process on Σ, the number of singular edges keeps growing but the number of singular leaves stays bounded, and for every non simply-connected leaf, K(L) remains unchanged. Therefore, after sufficiently long iteration of the pruning process, there is a boundary of a band e which does not belong to K(L) for any singular leaf L.

If δ is small enough, the part of the band removed (b − δ, b] × (0, 1) does not meet K(L) for any non simply-connected leaf L. Thus, properties (i) and (ii) hold if we choose for L′_{i} the Σ_{δ}-leaf containing K(L_{i}).

To handle property (iii), we take extra care in the choice of e and δ (but no extra pruning will be necessary). Given a singular leaf L, consider the minimal connected graph K′(L) ⊂ L containing K(L) and every singular edge in L. We choose e so that all singular edges of L_{e} \ e are on the same component of K′(L_{e}) \ e. We denote by L′_{e} the Σ_{δ}-leaf containing all the singular edges of K′(L_{e}) \ e. We choose δ small enough so that for every singular leaf L ≠ L_{e}, K′(L) ⊂ Σ_{δ}, and K′(L_{e}) \ e ⊂ Σ_{δ}. Moreover, we choose δ so that the new boundary e_{δ} = {b − δ} × [0, 1] of the narrowed band is in a singular leaf of Σ (this still enables us to choose δ as small as we want since the union of singular leaves is dense in Σ [Gab96]). We denote by L_{e_{δ}} the Σ_{δ}-leaf containing e_{δ}.

Consider a singular leaf L ≠ L_{e}, and let L′ be the Σ_{δ}-leaf containing K′(L). Recall that DL is the graph of directions of L and that q : DL → L is the natural projection. Every connected component of DL contains a point whose projection in L is the endpoint of a singular edge. Thus, q^{−1}(K′(L)) meets every connected component of DL. Therefore, if L ≠ L_{e}, then C_{Σ_{δ}}(L′) ≥ C_{Σ}(L).

Similarly, one has C(L′_{e}) ≥ C(L_{e}) − 1: this is because the preimage in DL_{e} of the component of K′(L′_{e}) \ e containing all the singular edges distinct from e meets all the connected components of DL but one. This gives the inequality C(Σ_{δ}) ≥ C(Σ) − 1.

Now remember that we chose δ so that e_{δ} lies in a singular leaf L of Σ, and look at the leaf L_{e_{δ}}. If L ≠ L_{e}, then we get C_{Σ_{δ}}(L_{e_{δ}}) ≥ C_{Σ}(L) + 1 because the new singular edge e_{δ} disconnects a component of DL. Similarly, if L = L_{e}, we get C_{Σ_{δ}}(L_{e_{δ}}) ≥ C_{Σ}(L_{e}). Thus, on the whole, C(Σ_{δ}) ≥ C(Σ) and (iii) holds.

It is not quite true that Σ_{δ} has compact leaves but it is proved in [Gui98] that the only minimal components that may appear in Σ_{δ} are exotic and that performing such a band narrowing decreases the number of ends of singular leaves. Therefore, after a finite sequence of pruning and narrowing operations, we get a foliated 2-complex Σ′ with compact leaves (see [Gui98]). This proves the lemma.

6 Complexity of surface components

Proposition 6.1. Consider (T, Γ, H) an almost geometric action of a pair with a nice decomposition as a graph of actions G (definition 3.5).

Then, for each surface type vertex action (T_{v}, Γ_{v}^{0}), there is a simplicial action (T′_{v}, Γ_{v}^{0}) with cyclic arc stabilizer, in which every point stabilizer of (T_{v}, Γ_{v}^{0}) fix a point, and satisfying

C(T_{v}) ≤ 3C(T′_{v}) + 6.

There is a graph of actions G′ obtained from G by changing T_{v} to T′_{v} such that (T_{G′}, Γ, H) is an almost geometric action with no exotic or surface component, arc stabilizers are in C if those of T are, and satisfying

C(T) ≤ C(T_{G′}) + 10γ.

Proof. Unlike in the exotic case, we have now that for every length-0 edge e, its edge group Γ_{e} in G is in C. This is because point stabilizers in a surface-type or homogeneous action (T_{v}, Γ_{v}^{0}) are cyclic (at most ℤ/2 in the axial case) and C is closed under cyclic extension. Therefore, G has at most γ reduced vertices and hence at most γ surface-type and homogeneous components.

Let v be a vertex such that T_{v} is of surface type, let Σ_{v} be the corresponding foliated surface, and ρ_{v} : π_{1}(Σ_{v}) → Γ_{v}^{0} defining the cover of Σ_{v}. Γ_{0} is the fundamental group of the orbifold with boundary obtained from Σ_{v} by collapsing a boundary component B to a conic point of angle 2π/n where n is the index of π_{1}(B) ∩ ker ρ_{v} in π_{1}(B) (when n = ∞, don’t collapse B). We rather consider the surface Σ′_{v} obtained from Σ_{v} by collapsing to a (regular) point each boundary components with n = 1. Each interior singularity s of the foliation induced on Σ′_{v} corresponds to such a collapse. Because of the tameness, it is a p(s)-pronged singularity with p(s) ≥ 2. When p(s) = 2, it is a false singularity. For boundary singularities, p(s) will denote the number of branches going into the interior of Σ_{v} so that p(s) = 0 for regular points on the boundary (see [FLP79] for background on measured foliations on surfaces).

Now let’s compute C(T_{v}) using lemma 4.1. A singular leaf L in Σ_{v} is composed of a circle made of k singular edges, and of a semiline made of regular edges coming out of each vertex which does not lie in ∂D. Thus DL has C(L) = k − #(L ∩ ∂D) connected components. Thus, all those complexities add up to 2#{bands of Σ_{v}}−#∂D = −2χ(Σ_{v}). But some of these leaves correspond to non-reduced points: this occurs for each singular leaf L with complexity 2 such that π_{1}(L) ⊂ ker ρ_{v}, in other words for false interior singularities of Σ′_{v}. Hence,

C(T_{v}) = −2χ(Σ_{v}) − 2#{false interior singularities of Σ′_{v}}
= −2χ(Σ′_{v}) + 2#{true interior singularities of Σ′_{v}}

On the other hand, a standard Euler characteristic argument shows that

−2χ(Σ′_{v}) = ∑_{s interior singularity of Σ′_{v}}[p(s) − 2] + ∑_{s boundary singularity of Σ′_{v}} p(s)

≥ #{true interior singularities of Σ′_{v}}

Thus, C(T_{v}) ≤ −6χ(Σ′_{v}).

Consider on Σ′_{v} a maximal system of disjoint, one-sided, simple closed curves. Cutting Σ′_{v} along those curves gives an orientable surface with same Euler characteristic as Σ′_{v}. Decompose it into −χ(Σ′_{v}) pants. The boundaries of these pants give at least −χ(Σ′_{v}) − 1 not boundary-parallel disjoint curves in Σ′_{v}. Each vertex of the action (T′_{v}, Γ_{v}^{0}) dual to this system of curves has a non-cyclic stabilizer, so must be reduced. Thus C(T′_{v}) ≥ −2χ(Σ′_{v}) − 2. Therefore, we get the expected inequality C(T_{v}) ≤ 3C(T′_{v}) + 6.

Let G′ be the graph of actions obtained by replacing the vertex tree T_{v} by T′_{v} for each surface type vertex action T_{v} (this is possible because fundamental groups of boundary components of Σ′_{v} fix a point in T′_{v}). In view of corollary 2.3, we get

C(T) ≤ C(T_{G′}) + ∑_{T_{v} surface type} (C(T_{v}) − C(T′_{v})) + 2γ
≤ C(T_{G′}) + ∑_{T_{v} surface type} (2C(T′_{v}) + 6) + 2γ

If we replace homogeneous actions of G′ by trivial actions, the obtained simplicial action of Γ has complexity at least ∑_{T_{v} surface type} C(T′_{v}) (use lemma 2.2 where one can take a = 0 since no vertex of T′_{v} has a cyclic stabilizer). Thus ∑_{T_{v} surface type} C(T′_{v}) ≤ γ. Since the number of surface type vertices is at most γ, we get C(T) ≤ C(T_{G′}) + 10γ.

7 Complexity of homogeneous components

Proposition 7.1. Let (T, Γ, H) be an almost geometric action of a pair having a nice decomposition as a graph of actions G with no exotic or surface component. Assume that arc stabilizers of T are in C.

Then C(T) ≤ 3γ + γ2^{γ+dim H_{1}(Γ;ℤ/2)−1}

Proof. Like in proposition 6.1, for every edge e of G, its edge group Γ_{e} in G is in C (even if e has length 0). Therefore, the number of homogeneous components in G is bounded by γ. If we have a bound B for the complexity of a homogeneous component occurring in G, we deduce from lemma 2.2 that

C(T) ≤ C(G̃_{0}) + 2a + ∑_{v} C(T_{v}) ≤ 3γ + γB.

Let v be a vertex corresponding to a homogeneous component in G. Since T_{v} is a line, C(T_{v}) = 0 when v is an orientable homogeneous component. When v is non-orientable, then Γ_{v}^{0} ≃ 𝔻_{n} acts faithfully on ℝ as a subgroup of Isom (ℝ) generated by a reflexion and n rationally independent translations. Since T_{v} has complexity 2^{n}, we just have to bound n.

Let G′ be the graph of groups obtained from the graph of groups underlying G by killing edge groups and let Γ′ = π_{1}(G′). Γ maps onto Γ′ which maps onto Γ′_{v} (Γ′_{v} is the vertex group of v in G′). Let C be the subspace of H_{1}(Γ_{v}^{0}; ℤ/2) generated by the images of the edge groups of edges incident on v. Note that dim C ≤ γ since images of edge groups in Γ_{v}^{0} are cyclic. We have H_{1}(Γ′_{v}; ℤ/2) = H_{1}(Γ_{v}^{0}; ℤ/2)/C and

n + 1 = dim H_{1}(Γ_{v}^{0}; ℤ/2) = dim H_{1}(Γ′_{v}; ℤ/2) + dim C ≤ dim H_{1}(Γ; ℤ/2) + γ.

Therefore, C(T_{v}) is bounded by B = 2^{dim H_{1}(Γ;ℤ/2)+γ−1}.

8 Conclusion

Here, we put everything together to prove the Theorem. Consider a finitely presented pair (Γ, H). Let C be a class of subgroups of Γ stable by taking subgroups and cyclic extensions such that (Γ, H) satisfies the accessibility condition with respect to C with accessibility constant γ. Consider a stable action (T, Γ, H) with arc stabilizers in C.

By lemma 3.4, (T, Γ, H) is a strong limit of almost geometric actions (T_{i}, Γ, H) having a nice decomposition. By Proposition 3.1, C′(T) ≤ lim inf C(T_{i}).

We now fix an index i and we bound C(T_{i}). Propositions 5.1 and 6.1 provide from T_{i} an action (T′_{i}, Γ, H) without exotic components or surface components with arc stabilizers in C such that C(T_{i}) ≤ C(T′_{i}) + 12γ. Proposition 7.1 concludes that C(T′_{i}) ≤ 3γ + 2^{γ+dim H_{1}(Γ;ℤ/2)−1}.

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Institut Fourier, UMR 5582, BP74, Université Grenoble 1, 38402 Saint-Martin d’Hères Cédex, France.

e-mail: vincent.guirardel@ujf-grenoble.fr