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Hall's Theorem for limit groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela.

view limit groups are precisely the finitely generated groups with the same existential theory as a free group [17]. In [20,21,23,22,24,26,25], Zlil Sela solved a famous problem of Alfred Tarski by classifying the elementarily free groups, those limit groups with the same elementary theory as a free group. O. Kharlampovich and A. Miasnikov also announced a solution to Tarski's Problem in [11,12,14,13,15].
Work on Tarski's Problem has led to the development of a powerful structure theory for limit groups. One form of this is given by a theorem of Kharlampovich and Miasnikov [12] (see also [4]). Let ICE be the smallest set of groups containing all finitely generated free groups that is closed under extending centralizers. Limit groups are precisely the finitely generated subgroups of groups in ICE . We make extensive use of this characterization.
Marshall Hall Jr. proved in [7] that every finitely generated subgroup of a finitely generated free group is a free factor in a finite-index subgroup, and that this finite-index subgroup can be chosen to exclude any element not in the original subgroup. As a consequence, finitely generated free groups are subgroup separable (also known as LERF ); that is, every finitely generated subgroup is closed in the profinite topology. This is a strong algebraic condition that implies, for instance, that the generalized word problem is solvable. In the 1970s it became apparent that subgroup separability has a natural topological interpretation. Peter Scott used hyperbolic geometry in [18] to prove that surface groups are subgroup separable, while J. R. Stallings exploited the topology of graphs to reprove Hall's Theorem and other properties of free groups in [29]. Stallings' techniques have been extended by, among others, Rita Gitik [6], who proved that, in certain circumstances, the amalgamated product of a subgroup separable group with a free group over a cyclic subgroup is subgroup separable, and D. T. Wise [33], who classified the subgroup separable graphs of free groups with cyclic edge groups. Sela [27] asked if limit groups are subgroup separable. In [30], the author answered Sela's question in the affirmative for elementarily free groups. Here we extend that result.
Theorem A (Corollary 3.9) Limit groups are subgroup separable.
It follows that limit groups have solvable generalized word problem. We also prove the stronger theorem that ICE groups are coset separable with respect to vertex groups.
Another strong consequence of Hall's Theorem is that every finitely generated subgroup H of a free group F is a virtual retract; that is, H is a retract of some finite-index subgroup of F . This property is referred to as local retractions in [16], in which connections between virtual retracts and 3-manifold topology are explored. Combining Theorem A with Theorem 3.1 of [2] it follows that every cyclic subgroup of a limit group is a virtual retract. We show here that limit groups virtually retract onto all their finitely generated subgroups.
Theorem B (Corollary 3.10) Let G be a limit group and H a finitely generated subgroup of G. Then there exists a finite-index subgroup K of G containing H and a retraction K → H.
As a corollary one obtains a purely topological proof that finitely generated subgroups of limit groups are quasi-isometrically embedded. See subsection 3.5 for more details.
The paper is organized as follows. In section 1 the notions of subgroup separability and local retractions are introduced. An example is given of how Stallings' techniques can be used to prove a strong theorem about finitely generated free groups. In order to extend these ideas, we need graphs of spaces in the spirit of [19]. We briefly outline some salient features of the theory of limit groups and their structure. In section 2 we develop the language of elevations (used by Wise in [33]) and pre-covers (analogous to those used by Gitik in [6]). Using these we can state a precise generalization of Stallings' ideas to the context of graphs of spaces (Theorem 2.21), which we call Stallings' Principle. There are still substantial obstacles to applying this to limit groups, and section 3 is devoted to overcoming them. Specifically, we define tame coverings, in which we have strong control over elevations of loops. The main technical result, Theorem 3.8, asserts that coverings of K(G, 1)s for groups in ICE are tame over certain sets of loops. Theorems A and B follow quickly.
This paper makes use of the results about pre-covers and elevations in [30], whither we often refer the reader for proofs. However, it should be emphasized that Theorems A and B are independent of the main theorem of [30] (Theorem 5.13). Theorem A was announced in [32], with a citation of an earlier version of this paper entitled 'Limit Groups are Subgroup Separable'.
Throughout, if g and h are group elements then g h denotes h −1 gh.

Acknowledgements
I would like to thank William Dison, Andrew Harkins and Michael Tweedale for patiently listening to rehearsals of the ideas of this work, Alan Reid for drawing local retractions to my attention, and particularly Martin Bridson for his many insightful comments and suggestions.
1 Fundamental notions 1.1 Separability of subgroups and double cosets Note that if G is subgroup separable and H ⊂ G is a subgroup then H is also subgroup separable.
A group G has solvable generalized word problem if there exists an algorithm that, given an element g ∈ G and a finite subset S ⊂ G, determines whether or not g ∈ S . Just as residually finite groups have solvable word problem, so subgroup separable groups have solvable generalized word problem.
We will also consider the separability of double cosets. Subgroup separability has a beautiful topological reformulation. We work in the category of combinatorial complexes and combinatorial maps. If G is the fundamental group of a complex X then the separability of the subgroup H can be reformulated in terms of a condition on the covering of X corresponding to H. Lemma 1.3 (Scott's Criterion) Let X be a connected complex, H a subgroup of π 1 (X) and p : X H → X the (connected) covering corresponding to H. The following are equivalent. Proof. Fix basepoints x 0 ∈ X and x 0 ∈ p −1 (x 0 ). Suppose H is separable and ∆ ⊂ X H is a finite subcomplex. For each cell σ ⊂ ∆ fix a choice of barycentre a σ ∈ σ, in such a way that if cells σ and σ have the same image in X then p(a σ ) = p(a σ ). Choose paths α σ in X H from x 0 to a σ and denote byᾱ σ the reverse path from a σ to x 0 . For each distinct pair of cells σ, σ ⊂ ∆ with p(σ) = p(σ ) the concatenation of paths γ σ,σ = p • α σ · p •ᾱ σ lies in π 1 (X, x 0 ) but not in H. Since H is separable, for each such pair σ, σ there exists a finite-index subgroup K σ,σ ⊂ π 1 (X) with H ⊂ K σ,σ but g σ,σ / ∈ K σ,σ . Define a finite-sheeted coveringX → X by where the intersection ranges over each all pairs of cells σ, σ ⊂ ∆ with p(σ) = p(σ ). Then ∆ embeds inX by construction.
Conversely, suppose every finite subcomplex of X H embeds in a finitesheeted cover and let g ∈ π 1 (X) H. Fix a combinatorial loop representing g and consider its based lift g : [0, 1] → X H with g (0) = x 0 . Since g / ∈ H, g (1) = x 0 . By the supposition, there exists a finite-sheeted intermediate coveringX such that the image of g embeds inX. In particular, g / ∈ π 1 (X). Hence, H is separable.

Local retractions
A group G retracts onto a subgroup H if the inclusion map H → G has a left-inverse ρ. In this case the map ρ is called a retraction and the subgroup H is called a retract.

Definition 1.4 ([16])
A group G virtually retracts onto a subgroup H if there exists a finite-index subgroup K ⊂ G containing H and a retraction K → H. If G virtually retracts onto all its finitely generated subgroups then G is said to have local retractions or property LR.
Finitely generated abelian groups have local retractions. Hall's Theorem asserts that every finitely generated subgroup H of a free group F is a free factor in a subgroup K ⊂ F of finite index; in particular, H is a retract of K, so F has local retractions. Notice that having local retractions passes to subgroups.
Having local retractions is a strong property, with many deep consequences. For more information, see [16]. We will mention just one such consequence. Recall that a map of metric spaces f : X → Y is a quasi-isometric embedding if there exist constants λ and such that, for all x 1 , x 2 ∈ X, A finitely generated subgroup H of a finitely generated group G is quasiisometrically embedded if, for some choice of finite generating sets and associated word metrics, the inclusion map H → G is a quasi-isometric embedding. This is in fact independent of the choices of generating sets. Since the word metric on a group is only well-defined up to quasi-isometry, this is a natural notion of good geometric behaviour for a subgroup. Lemma 1.5 If a finitely generated group G virtually retracts onto a finitely generated subgroup H then H is quasi-isometrically embedded.
Proof. There exists finite-index K ⊂ G and a retraction ρ : K → H. Since K decomposes as H ker ρ there is a finite generating set for K of the form S ∪ T where H = S and T ⊂ ker ρ. In the word metric with respect to this generating set, the inclusion H ⊂ K is an isometric embedding. But K → G is a quasi-isometry, so H → G is a quasi-isometric embedding.

The motivating idea
The proof of our main technical result, Theorem 3.8, is inspired by Stallings' famous proof of Hall's Theorem [29] which we outline here. Theorem 1.6 Let X → X be an immersion of finite graphs. Then X embeds in some finite graphX and there exists a covering mapX → X extending X → X.
We postpone the proof of this to subsection 2.5. Proof. Let F be a finitely generated free group and H a finitely generated subgroup. Realize F as the fundamental group of some finite graph X, in which case H corresponds to a covering X H → X. Let ∆ be a finite subgraph of X H . Since H is finitely generated, there exists a finite connected subgraph X ⊂ X H so that the inclusion map is a π 1 -isomorphism. Enlarging X if necessary, it can be assumed that ∆ ⊂ X . The restriction of the covering map X → X is an immersion of finite graphs. By Theorem 1.6, X can be completed to a finite-sheeted coverX into which ∆ embeds. Since X is a subgraph ofX, H is a free factor and, in particular, a retract of π 1 (X).
We aim to prove a version of Theorem 1.6 in the context of graphs of spaces (or, equivalently, graphs of groups-see subsection 1.4). Three general theorems have been proved about subgroup separability of graphs of groups. One concerns the amalgamation of a subgroup separable group and a free group [6]. Another concerns graphs of free groups [33]. In [30], we proved that elementarily free groups are subgroup separable. (Elementarily free groups can be constructed inductively by gluing surfaces along their boundaries.) In all these theorems, at least one vertex group of the graph of groups is free. Free groups play such a key role because Theorem 1.6 provides great flexibility in constructing finite-index subgroups of free groups with special properties.
The flexibility required in [6], [30] and [33] is similar to the following corollary of Theorem 1.6. A collection g 1 , . . . , g n of elements of a group G is independent if, whenever there exists h ∈ G such that g h i and g j commute, then in fact i = j. Corollary 1.8 Let X be a graph and let H ⊂ π 1 (X) be finitely generated. Let X H → X be the covering corresponding to H and ∆ ⊂ X H a finite subcomplex. Let {γ i } be a finite independent set of elements of π 1 (X) that each generate a maximal cyclic subgroup. For each γ i , consider a finite collection Note that Corollary 1.8 implies Corollary 1.7.
Proof. Fix a basepoint x ∈ X and let (X,x) → (X, x) be the universal covering. Since H is finitely generated there exists some connected finite subgraph X ⊂ X H such that the inclusion map is a π 1 -isomorphism. Enlarging X we can assume that it contains ∆ and the basepoint x ∈ X H . Note that X H X is a forest.
Denote the universal covering map R → S 1 by p. Identifying each loop γ g j i with a based map (S 1 , p(0)) → X, each concatenation γ g j i • p lifts to a based map γ H j : (R, 0) → (X H , x ) and further toγ j : (R, 0) → (X,x), where we have dropped the i's for notational convenience. Since γ g j i ∩H = 1 the γ H j are proper. Eachγ j contains Axis(γ g j i ) in its image . If the intersection of the images of γ H j and γ H j contains an infinite ray then, for some h ∈ H, the intersection contains an infinite ray. Therefore, for some integers m and n, the element fixes a point inX and so is trivial, whence i = i and j = j . Enlarging X still further we can also assume that, for all sufficiently large d: 1. each γ H j restricts to an arc of combinatorial length d times the combinatorial length of γ i ; 2. for any distinct γ By identifying each γ j (a j ) with γ j (b j ) we obtain an immersionX → X such that γ g j i ∩ π 1 (X) = (γ g j i ) d for every i and j. Applying Theorem 1.6 toX gives the required coverX. Note that X is a subgraph ofX so H is a free factor of π 1 (X) and, in particular, a retract.
In the terminology of section 3, Corollary 1.8 asserts that the cover X H is tame over the set {γ i }. The lifts γ H j are elevations and X is a pre-cover. The proof of Corollary 1.8 makes use of the fact that the action of π 1 (X) on the treeX is free. In extending these ideas to limit groups, the key observation is that a similar construction can be made when we make the weaker assumption that an action on a tree is acylindrical (see Lemma 2.24).

Graphs of spaces
The fundamental groups of graphs are always free groups. The groups we consider will usually not be free, so we need a more general notion. A graph of spaces Γ consists of: 1. a set V (Γ) of connected spaces, called vertex spaces; 2. a set E(Γ) of connected spaces, called edge spaces; When the edge in question is unambiguous, we often suppress the superscript and refer to ∂ e ± simply as ∂ ± .
The associated topological space |Γ| is defined as the quotient of obtained by identifying (x, ±1) with ∂ e ± (x) for each edge space e and every x ∈ e. We will usually assume that |Γ| is connected. If X = |Γ| we will often say that Γ is a graph-of-spaces decomposition for X, or just that X is a graph of spaces. The underlying graph of Γ is the abstract graph given by replacing every vertex and edge space of Γ by a point. If X is the topological space associated to the graph of spaces Γ, the underlying graph is denoted Γ X . Note that there is a natural π 1 -surjective map φ X : X → Γ X .
Given a graph of spaces X, consider a subgraph Γ ⊂ Γ X . The corresponding graph of spaces X = φ −1 X (Γ ) has a natural inclusion X → X and is called a sub-graph of spaces of X. If X and Y are graphs of spaces, a map Definition 1.9 A connected sub-graph of spaces X ⊂ X with finite underlying graph such that the inclusion map is a π 1 -isomorphism is called a core for X.
Note that if π 1 (X) is finitely generated then a core always exists.
The fundamental group of a graph of spaces is naturally a graph of groups by the Seifert-van Kampen Theorem. For more on graphs of spaces, graphs of groups and Bass-Serre Theory see [19] and [28]. Our notation for graphs of spaces is different to that of [19].

Limit groups
Fix a finitely generated non-abelian free group F of rank at least 2.
Likewise, G is fully residually free (or ω-residually free) if, for any finite subset S ⊂ G, there exists a homomorphism f : G → F such that f | S is injective. A finitely generated, fully residually free group is called a limit group.
Sela's original definition of limit groups is different, but the two definitions are equivalent (see [20], Definition 1.2 and Theorem 4.6). We will need three related properties of limit groups. A group is CSA (standing for conjugately separated abelian) if every maximal abelian subgroup is malnormal. Definition 1.11 Let Γ be a graph-of-groups decomposition for a group G, so G acts on the Bass-Serre tree T . Endow T with a geodesic metric in which every edge has length 1. Then Γ is k-acylindrical if, for every g ∈ G 1, the set of fixed points of the action of g on T is either empty or has diameter at most k. If Γ is k-acylindrical for some finite k then Γ is called acylindrical. Lemma 1.12 Let G be a limit group and consider a non-trivial decomposition Γ of G as an amalgamated free product G = A * C B with C abelian.

ICE spaces
There is a powerful structure theory for limit groups. Let G be a group and Z ⊂ G the centralizer of an element. Then the group G = G * Z (Z × Z n ) is said to be obtained from G by extension of a centralizer. We call Z × Z n the extended centralizer.
Definition 1.13 Let ICE (standing for iterated centralizer extension) be the smallest class of groups containing all finitely generated free groups that is closed under extension of centralizers. A group in ICE is called an ICE group.
All ICE groups are limit groups (see, for example, Theorem 4.15 of [31]).
Remark 1.14 Suppose G is obtained from G ∈ ICE by extensions of a centralizer and the centralizer Z is non-cyclic. By assertion 1 of Lemma 1.12, Z is abelian. By assertion 2 of Lemma 1.12 and induction, Z is conjugate to some previous extended centralizer Z × Z m . It follows that, in the construction of ICE, it can always be assumed that Z is cyclic.
Every G ∈ ICE has a natural finite K(G, 1), constructed as follows. If G is free then K(G, 1) can be taken to be a compact graph of suitable rank. If G is obtained from G by extension of a (cyclic) centralizer and Y = K(G , 1) then given an essential closed curve ∂ + : S 1 → Y representing a generator of Z and a coordinate circle . We call the resulting class of spaces ICE spaces. Note that each ICE space X has an associated graph-of-spaces decomposition with vertex spaces Y and T and edge space a circle.
The importance of the class ICE lies in the following theorem of O. Kharlampovich and A. Miasnikov.

Theorem 1.15 ([12]) A group is a limit group if and only if it is a finitely generated subgroup of an ICE group.
Another proof using different techniques is given in [4].
Our strategy to prove Theorems A and B is to prove them by induction for all ICE groups, since subgroup separability and local retractions both pass to subgroups. As mentioned in subsection 1.3, all previous combination theorems for subgroup separability rely on some of the vertex groups being free. This is not true of ICE groups. Therefore we will need to prove a strong result, analogous to Corollary 1.8, for all groups in ICE.

Elevations and pre-covers 2.1 Elevations
Let X be a graph of spaces and X → X a covering map. Then X inherits a graph-of-spaces decomposition. We want to understand the edge maps of X . To do this we study elevations, natural generalizations of lifts that were introduced by Wise in [33].
Two elevations f 1 : A simple example of some elevations is illustrated in Figure 1. In practice, we will often abuse notation and refer to just the lift f as an elevation of f .

Definition 2.2 (Degree of an elevation)
The degree of the elevation f : In the light of Remark 2.3, it makes sense to consider elevations of a map defined up to free homotopy.

Remark 2.4
If X has a graph-of-spaces decomposition Γ and X → X is a covering space then X inherits a graph-of-spaces decomposition Γ X , with vertex spaces the connected components of the pre-images of the vertex spaces of X and edge spaces and maps given by all the (isomorphism classes of ) elevations of the edge maps to the vertex spaces of X .
Elevations can also be understood algebraically, in terms of fundamental groups.
. The composition f • p admits a lift f : A → B to B , which is an elevation of f . Every elevation of f to B arises in this way.
The proof is standard covering-space theory. See, for example, Proposition 1.33 of [8].
Fix g ∈ π 1 (B, b). The terminus of the lift at b of g to B gives a welldefined new basepoint β ∈ B , and so defines an elevation f : When do two elements of π 1 (B, b) yield isomorphic elevations? Lemma 2.7 (Elevations correspond to double cosets) Consider the situation of Lemma 2.6. Let g 1 , g 2 ∈ π 1 (B, b) and let f i : Then f 1 and f 2 are isomorphic elevations if and only if A, a)).
Proof. Suppose f 1 and f 2 are isomorphic, so there exists a covering transformation ι : . Fix representative loops for g 1 and g 2 , and let g 1 and g 2 be their respective lifts at b to B . The concatenation g 2 · f 2 • γ · (g 1 ) −1 makes sense and determines an element of as required. Now suppose that Then there exists γ ∈ π 1 (A, a) such that g 2 f * (γ)g −1 1 ∈ π 1 (B , b ) and so Let γ be the lift at α 2 of γ to A 2 . By standard covering space theory, there exists a covering transformation ι : Henceforth, we will often suppress mention of basepoints. To summarize the above discussion, the (isomorphism classes of) elevations of a map f : A → B to a cover B are identified with the set of double cosets π 1 (B )\π 1 (B)/(f * π 1 (A)).  2. f * descends to a monomorphism f * : π 1 (A)/π 1 (Ã) → π 1 (B)/π 1 (B); 3. andf is π 1 (A)-equivariant-that is, for g ∈ π 1 (A), where τ g denotes the covering transformation by which g acts onÃ and, likewise,f * (g) acts onB by τf * (g) . Proof.

Axes of translation
As remarked in subsection 1.4, a graph of spaces X is equipped with a map to the underlying graph φ X : X → Γ X . One makes Γ X into a graph of groups by labelling each vertex and edge with the fundamental group of its pre-image. IfX → X is the universal covering then the underlying graph ΓX is naturally identified with the Bass-Serre tree of Γ X and the map φX :X → ΓX is π 1 (X)equivariant. (There is a subtlety here to do with basepoints. The action oñ X and the action on ΓX both depend on choices of basepoints. The map φX is equivariant if it maps the basepoint ofX to the basepoint of ΓX.) More generally, if X H → X corresponds to the subgroup H ⊂ π 1 (X) then the underlying graph Γ X H is naturally identified with the quotient graph H\ΓX.
It is a fundamental fact about the action of a group G on a tree that every g ∈ G either fixes a point or acts by translating an embedded line, denoted Axis(g) (see, for instance, [28]). Elements of the first sort are called elliptic and elements of the second sort are called hyperbolic. Accordingly, a loop δ : S 1 → X is called elliptic if it is freely homotopic into a vertex space; otherwise, it is called hyperbolic.
Proof. Let τ g be the covering transformation ofX corresponding to g. Sincẽ δ is the elevation of δ at the terminus of the lift of g atx 0 it follows that τ −1 g •δ is the elevation of δ atx 0 . By Lemma 2.10, therefore, τ −1 g •δ is a δ-equivariant map. Henceδ is a δ (g −1 ) -equivariant map.
As φX :X → ΓX is also equivariant we have that φX It is a fact about group actions on trees that any subtree invariant under the action of a hyperbolic element contains the axis and any subtree invariant under the action of an elliptic element contains a fixed point.

Acylindrical graphs of groups and proper loops
The fact that an ICE group decomposes as an acylindrical graph of groups has profound implications for the structure of limit groups. Definition 2.12 Let X be a graph of spaces and consider an arbitrary covering X → X with π 1 (X ) finitely generated; let φ : X → Γ X be the map to the underlying graph. A loop γ : C ∼ = S 1 → X is called proper if, for any such covering X and any elevation γ of γ to X , the composition map λ = φ • γ : C → Γ X is a proper map.
In the case when X really is a graph, all loops are proper.
Lemma 2.13 Let X be a compact graph. Then any loop γ : C → X is proper.
Proof. Let X → X be a covering with H = π ( X ) finitely generated and let γ : C → X be an elevation. If C is compact then there is nothing to prove, so assume that C ∼ = R. LetX be the universal cover of X and letX H be the minimal H-invariant subtree. If γ is not proper then Axis(γ) intersectsX H in infinitely many vertices and therefore an infinite ray. Consider a vertex v ofX H . There are only finitely many H-orbits of vertices inX H so, for some integer n and some element h ∈ H, γ n h −1 fixes v. As the action onX is free it follows that γ n ∈ H, so C is compact, contradicting the assumption that C ∼ = R.
This lemma is implicit in the proof of Corollary 1.8. We will need a similar result for ICE spaces, which we will prove by induction.
Lemma 2.14 Let Γ be a k-acylindrical graph of groups and consider the action of G = π 1 (Γ) on the Bass-Serre treeΓ. Suppose that γ ∈ G is a hyperbolic element with translation length |γ|. Let g be another element of G. If the intersection Axis(γ) ∩ gAxis(γ) is of length greater than k + |γ| then g commutes with γ. In particular, gAxis(γ) = Axis(γ).
We use this fundamental fact in the following technical lemma. For points a, b ∈Γ, denote by [a, b] the unique geodesic arc from a to b.

Lemma 2.15
Let X be a graph of spaces for which the corresponding graph of groups is k-acylindrical. Let γ ∈ π 1 (X) be a proper loop. Then for any covering X → X with finitely generated fundamental group and for any non-trivial g ∈ π 1 (X), there are at most finitely many cosets γ n π 1 (X ) in π 1 (X)/π 1 (X ) such that γ −n gγ n ∈ π 1 (X ).
Proof. Let G = π 1 (X) and let H = π 1 (X ). Without loss of generality, g ∈ H. If γ ∩ π 1 (X ) = 1 then there is nothing to prove, so assume that γ ∩ π 1 (X ) = 1. LetΓ be the Bass-Serre tree of the induced splitting of G equipped with the standard combinatorial metric d and letΓ H be the minimal H-invariant subtree. Since γ is proper,Γ H ∩ Axis(γ) is a compact arc α. Any element h ∈ H that commutes with γ must fix α, and so the whole of Axis(γ), pointwise. So, as the splitting is acylindrical, any such h is trivial.
Fix a basepoint t 0 ∈ α. Let |γ| be the translation length of γ and let β be the subarc of Axis(γ) consisting of those points that are a distance at most k + |γ| fromΓ H . Let n be large enough that γ ±n (β) ∩Γ H = ∅.
Using this observation, we can prove the extremely useful fact that all hyperbolic loops in ICE spaces are proper. Lemma 2.16 (Hyperbolic loops in ICE spaces are proper) Let X be an ICE space. Every hyperbolic loop δ : C → X is proper.
Proof. As usual X is constructed by gluing a torus T to an ICE space Y of lower level. By induction on level it can be assumed that the lemma holds for Y . Without loss of generality take δ to be based. The graph-ofspaces decomposition Γ for X induces a graph-of-groups decomposition of π 1 (X) which, abusing notation, we also denote by Γ. Decompose δ in Γ as an irreducible word δ = a 0 b 1 a 1 . . . b n a n where a i ∈ π 1 (T ) and b i ∈ π 1 (Y ). Let X → X be a covering of X and let δ : C → X be an elevation of δ, as in Definition 2.12. If C is compact then there is nothing to prove, so assume C ∼ = R. Let π 1 (e) = z . If λ = φ • δ is not proper then for some vertex space Y of X and two elevations ∂ + 1,2 : e 1,2 → Y the line λ crosses Y from ∂ + 1 (e 1 ) to ∂ + 2 (e 2 ) in infinitely many places. Therefore, without loss of generality, for infinitely many indices p, non-zero indices q p and some b k , Since the elevations ∂ + 1,2 are proper by induction, it follows that [b k , z] = 1, contradicting the irreducibility of the decomposition of δ.

Pre-covers
Pre-covers fill the role played by immersions of graphs in Stallings' proof of Hall's Theorem. In [6], Gitik uses a notion of pre-cover; our pre-covers are analogous to hers.ΓΓ Figure 2: The diagram shows the covering spaceΓ from Figure 1. A finite subgraphΓ ofΓ is drawn with a thick line. The restriction of the covering mapΓ → Γ is a pre-covering. The canonical completion ofΓ isΓ, which can be constructed by attaching the forest drawn with thin lines. Definition 2.17 Let X and X be graphs of spaces (X is not assumed to be connected). A pre-covering is a locally injective map X → X that maps vertex spaces and edge spaces of X to vertex spaces and edge spaces of X respectively, and restricts to a covering on each vertex space and each edge space. Furthermore, for each edge space e of X mapping to an edge space e of X, the diagram of edge maps is required to commute. The domain X is called a pre-cover. An example of a pre-cover, in the context of graphs, is illustrated in Figure 2.
The pre-covering X → X is finite-sheeted if the pre-image of every point of X is finite.

Remark 2.18
All the edge maps of X are elevations of edge maps of X to the vertex spaces of X . An elevation of an edge map of X to a vertex space of X that is not (isomorphic to) an edge map of X is called hanging. If none of the elevations are hanging then X is in fact a cover.
Proposition 2.19 (Canonical completion of pre-covers) Let X be a graph of spaces and X → X a pre-covering with X connected. Then π 1 (X ) injects into π 1 (X) and, furthermore, there exists a unique embedding X →X into a connected coveringX → X such that π 1 (X) = π 1 (X ) andX → X extends X → X. This is called the canonical completion of X → X.
For the proof, see Proposition 2.9 of [30]. In this situation, X is a core for X. It should be stressed that this construction does not preserve finiteness of sheets. Indeed, the completionX is finite-sheeted if and only if π 1 (X ) is of finite index in π 1 (X).ΓΓ  Note that A is a pre-cover of A since f is a map of graphs of spaces. If A is non-empty then is called an elevation of f to B .
When A → A is a genuine covering of A the elevation f is called full. In this case, it makes sense to define the degree of f to be the conjugacy class of π 1 (A ) in π 1 (A).
Note that this definition of an elevation to a pre-cover is slightly less general than the definition given in [30]. However, it is less complicated and will suffice for the purposes of this paper. Two examples of elevations to a pre-cover are illustrated in Figure 3.

Stallings' Principle
With the language of pre-covers and elevations we can generalize Theorem 1.6 to extend finite-sheeted pre-covers of graphs of spaces to finite-sheeted covers. We call this generalization Stallings' Principle. Let X be a graph of spaces and p :X → X a pre-covering. Consider an edge map ∂ : e → V of X and an elevation∂ :ē →V of ∂. Since the image of ∂ is contained in V and p restricts to a covering map onV it follows thatē → e is a covering sō ∂ is full. Indeed,∂ is a genuine elevation toV . It therefore makes sense to consider the degree of∂ (see Definition 2.20).

Theorem 2.21 (Stallings' Principle)
LetX → X be a pre-covering with the following property: for each edge space e of X with edge maps ∂ ± : e → V ± , for each conjugacy class D of subgroups of π 1 (e), there exists a bijection between the set of elevations of ∂ + toX of degree D and the set of elevations of ∂ − toX of degree D. ThenX → X can be extended to a coveringX → X, with the same set of vertex spaces asX.
For the proof, see Proposition 3.1 of [30]. The crucial feature of this construction (as opposed to, say, Proposition 2.19) is that it extends finitesheeted pre-covers to finite-sheeted covers: sinceX has the same vertex spaces asX, ifX → X is finite-sheeted then so isX → X. From Theorem 2.21 one deduces Theorem 1.6 as follows.
Proof of Theorem 1.6. The immersion of graphs X → X is a pre-covering. Let N be the maximum number of pre-images in X of a vertex of X. Create a new (disconnected) pre-coverX by adding disconnected vertices to X so that every vertex of X has N pre-images. Now every edge map of Γ has N elevations to the pre-cover, all of the same (trivial) degree; so, by Theorem 2.21,X → X can be extended to a coveringX → X.
The proof of Corollary 1.7 makes use of the fact that a covering of a graph that corresponds to a finitely generated subgroup has a finite core. In contrast, in the case of a graph of spaces X, a covering X H corresponding to finitely generated subgroup H ⊂ π 1 (X) has a core X with finite underlying graph, but in general X → X is not finite-sheeted. Most of the rest of this paper is devoted to addressing this difficulty.

Making elevations full
Let δ : S 1 → X be a loop in X andX → X a pre-covering. Letδ :S 1 →X be an elevation of δ toX. The key technical construction enables us, under the hypothesis of disparity, to completeδ to a full elevation-that is, an elevation whose domain is homeomorphic to S 1 or R. Indeed, we can do this for sets of elevations. First we fix some notation. LetX be the canonical completion ofX sō δ is a restriction of an elevationδ :S 1 →X. Given x ∈ ∂S 1 , the closure of the adjacent component ofS 1 S 1 can be identified with some subinterval of [0, ∞), with x identified with 0. For sufficiently small > 0,δ((0, )) is contained in a single edge space ofX, corresponding to a hanging elevation ∂ x : e x →X of some edge map ∂. Definition 2.22 (Disparity) Let {δ i : C i → X} be a set of loops. Let X → X be a pre-cover, as above. A collection of elevations where eachδ j is an elevation of some δ i , is disparate if: 1. jC j is compact; 2. the map from j ∂C j to the set of hanging elevations of edge maps given by x →∂ x is injective; and, 3. for each x ∈ j ∂C j , e x is simply connected.
If a path δ cannot be homotoped (relative to its endpoints) off any of the edge or vertex spaces of X that it intersects, then δ is called reduced. Clearly, any path is homotopic to a reduced path. Recall that an elevationδ j to a pre-cover is full if its domainC j is a covering space of C i . The next lemma asserts that, under the hypothesis of disparity, we can extend a pre-cover to make a collection of elevations of reduced loops full. Lemma 2.23 (Making elevations full) LetX → X be a pre-covering, let {δ i : C i → X} be a set of reduced loops and let {δ j :C j →X} be a disparate set of elevations toX. Suppose that, for each j,Ĉ j → C i is a finite-sheeted covering extendingC j → C i . Then there exists a pre-coverinĝ X → X extendingX → X so that eachδ j extends to a full elevation IfX has finite underlying graph, eachĈ j → C i is finite-sheeted and there are finitely manyδ j then the resultingX has finite underlying graph. Furthermore, π 1 (X) = π 1 (X) * F where F is some finitely generated free group.
The proof of Lemma 2.23 is the same as that of Proposition 2.13 in [30]. However, because of its fundamental importance to our argument, we repeat the proof here. Figure 4 illustrates the idea of Lemma 2.23 in the situation of Figure 3.

Proof.
Let D be the closure of a component ofĈ j C j . Without loss identify D ≡ [−1, 1]. Consider the canonical completionX → X ofX → X. There is a unique liftδ + j : D →X of δ i toX so thatδ + j (1) =δ j (1). Let > 0 be minimal such thatδ + j ( − 1) lies in a vertex space. Let X ⊂X be the pre-cover consisting ofX together with the vertex spaces and edge spaces containingδ + j (( − 1, 1)). Letẽ + be the edge space ofX so that δ + j ([−1, − 1]) ⊂ẽ + × [−1, 1] and without loss of generality assume that 1] for some unique edge spaceẽ − ofX. Without loss of generality, assumeδ − j (−1) ∈ẽ − × {−1}. The edge spacesẽ + andẽ − are both simply connected covers of some edge space e of X. There exists some unique covering transformation τ : Now letX be the pre-cover given by X together with the additional edge spaceẽ + ; the additional edge maps are ∂ẽ + + and ∂ẽ − − • τ . Since {δ j } is disparate, this can be done for every of component ofĈ j C j .

Ensuring disparity
In the light of Lemma 2.23, we will need to be able to guarantee that collections of elevations are disparate. We prove under certain assumptions that, in the underlying graph of the canonical completion, any pair of elevations shares at most finitely many vertices. Therefore, by enlarging our pre-cover to a larger core of the canonical completion, a finite set of elevations can be made disparate.
Lemma 2.24 (Making elevations disparate) Suppose X is an ICE space. Let {δ i : C i → X} be a finite, independent collection of hyperbolic loops such that each δ i generates a maximal abelian subgroup of π 1 (X). LetX be a connected pre-cover with finitely generated fundamental group and {δ j :C j →X} a finite collection of elevations of the δ i . Then there exists a pre-coverinĝ X → X extendingX → X such that: 1.X has finite underlying graph; 2. π 1 (X) = π 1 (X); 3. eachδ j extends to an elevationδ j toX; and 4. the collection {δ j } is disparate.
Proof. Let H = π 1 (X), and let T be the Bass-Serre tree of the induced splitting of π 1 (X), soΓ = H\T is the underlying graph of the canonical completionX ofX. Think ofΓ as a graph of groups. The underlying graph Γ ofX is a finite core forΓ. The Bass-Serre tree ofΓ is naturally identified with an H-invariant subtree T H of T .
The first observation is that, outside of a finite subgraph, all the edge and vertex stabilizers ofΓ are trivial. Otherwise, there are elements of π 1 (X) fixing arbitrarily large subtrees of T , contradicting acylindricality. Therefore, enlargingX inX, it can be assumed that every edge stabilizer inΓ Γ is trivial. As a consequence, H freely permutes the components of T T H .
Eachδ j is the restriction of some elevationδ j :C j →X which corresponds to a double coset Hg j δ i . Consider the natural map φX :X →Γ and set λ j = φX •δ j . By Lemma 2.11, the image of λ j contains the projection of the line l j = g j Axis(δ i ) ⊂ T toΓ.
Since λ j is proper (by Lemma 2.16), eachC j is compact wheneverΓ is a finite graph.
Note that ifδ j is infinite-degree then l j ∩ T H is finite, so there exist two distinct components Λ ± j of T T H with l j ∩ Λ ± j an infinite ray. We will now show that no two λ j share more than finitely many edges in their images. Suppose that the intersection of the images of λ 1 and λ 2 contains infinitely many edges. Thenδ 1 andδ 2 are infinite-degree elevations and the images of l 1 and l 2 inΓ share infinitely many edges, since H is finitely generated. Without loss of generality, there exist infinitely many h i ∈ H mapping an edge in Λ + 1 ∩ l 1 to an edge in Λ + 2 ∩ l 2 . Since H freely permutes the components of T T H these h i are all equal to some h ∈ H, and hl 1 ∩ l 2 contains infinitely many edges. By acylindricality δ hg 1 −1 1 and δ g −1 2 2 commute, so δ 1 = δ 2 and Soδ 1 andδ 2 are isomorphic elevations.
Finally, we also need to check that, for each infinite-degreeδ j , the limits lim t→∞ λ j and lim t→−∞ λ j differ-otherwise, for any finite core ofΓ, the two endpoints of the restriction ofδ j would always lie in the same vertex space. However, if the limits lim t→∞ λ j and lim t→−∞ λ j coincide then there exists h ∈ H mapping l j ∩ Λ + j to l j ∩ Λ − j , so by acylindricality h ∈ δ g j −1 i . This contradicts the assumption that the elevationsδ j are of infinite degree.
In summary,X can be enlarged to a larger coreX ⊂Γ satisfying the conclusions of the lemma.

.1 Tame covers
One might hope to prove Theorems A and B for all groups in ICE, and hence for all limit groups, by induction. However, neither subgroup separability nor local retractions is strong enough to serve as an inductive hypothesis. We need a still stronger property-tameness. Definition 3.1 Consider a complex X, a covering X → X and a finite (possibly empty) collection of independent, essential loops The cover X is tame over L if the following holds.
Let ∆ ⊂ X be a finite subcomplex and let be a finite collection of (pairwise non-isomorphic) infinite-degree elevations where each δ j is an elevation of some δ i ∈ L. Then for all sufficiently large positive integers d there exists an intermediate finite-sheeted covering X →X → X so that: 1. every δ j descends to an elevationδ j of degree d; 2. theδ j are pairwise non-isomorphic; 3. ∆ embeds intoX; 4. there is a retraction ρ : π 1 (X) → π 1 (X ) such that ρ(δ j * (π 1 (Ĉ j ))) ⊂ δ j * (π 1 (C j )) for each j.
Suppose that a covering X → X is tame over the empty set of loops in X. It follows, by Lemma 1.3, from property 3 of Definition 3.1 that π 1 (X ) is separable in π 1 (X). Likewise, it follows immediately from property 4 of Definition 3.1 that π 1 (X ) is a virtual retract of π 1 (X). We therefore have the following. Lemma 3.2 If a covering of a graph of spaces X → X is tame over the empty set of loops in X then π 1 (X ) is separable and a virtual retract in π 1 (X).
As a first example, covers of tori are tame over coordinate circles.

Lemma 3.3
If T is an n-torus, T → T is a covering and δ : S 1 → T is a coordinate circle then T is tame over the singleton {δ}.
The cover T can be taken to be the cylinder for some integers µ i , and the elevations δ can be taken to be the maps Consider a compact cylindrical subspace of the form If the a i are sufficiently negative and the b i are sufficiently positive then ∆ lies in the interior of Ω and, furthermore, Ω intersects each δ i in a non-trivial interval [a 1 , b 1 ]×{x i }. As long as d is sufficiently large we can take b 1 = a 1 + d. There is an action of Z k on T by covering transformations for which Ω is a fundamental domain; letT be the quotient. This is the required intermediate covering.
However, if L consists of two or more loops on T then the covering T → T need not be tame over L. This is what goes wrong in the example given in [3] of a 3-manifold group that is not subgroup separable.

Making pre-covers finite-sheeted
Tameness enables us to pass from pre-covers with finite underlying graph to finite-sheeted pre-covers.
Proposition 3.4 (Passing to finite-sheeted pre-covers) Let X be a graph of spaces in which every edge space is a circle, and let {δ i : C i → X} be a collection of elliptic loops confined to the vertex spaces of X. Let X → X be a pre-covering with finite underlying graph. Every vertex space V of X covers some vertex space V of X; assume that each V is tame over the union of the set of edge maps incident at V and the set of δ i whose images lie in V . Let ∆ ⊂ X be a finite subcomplex and {δ j : C j → X } a collection of infinite-degree elevations of the δ i . Then for all sufficiently large d there exists a finite-sheeted intermediate pre-covering X →X → X such that: 1. the {δ j } descend to distinct, full elevations {δ j :C j →X} so that, for each j,C j → C is a covering of degree d;
Proof. Enlarging ∆ it can be assumed that it contains every compact edge space of X . Let V be a vertex space of X covering the vertex V of X. Set ∆ V = V ∩ ∆. Consider the edge maps ∂ i : e i → V of X incident at V that are infinite-degree elevations of ∂ ± : e → V . By the hypothesis of tameness there exists an intermediate finite-sheeted covering V →V → V so that ∆ V embeds inV , each ∂ i descends to a degree d elevation of ∂ ± and each δ j with image in V descends to a degree d elevationδ j :C j → V . If d is sufficiently large then it can be taken to be the same d over all vertex spaces of X . LetX be the graph of spaces with the same underlying graph as X but in which each vertex space V is replaced by the correspondingV . If e is an edge space of X then the edge map descends to a finite-degree map∂ ± :ē ± →V . By constructionē + covers e with the same degree asē − so this information defines a finite-sheeted pre-coverX as required.
It is clear that ∆ embeds intoX. As for the retraction, π 1 (X) decomposes as a graph of groups, with the same underlying graph as the decomposition for π 1 (X ), and every vertex group of π 1 (X ) is a retract of the corresponding vertex group of π 1 (X). These retractions piece together to give the required retraction ρ : π 1 (X) → π 1 (X ).

Completing pre-covers to covers
Consider an ICE space X, which as usual is constructed by gluing a torus T to a simpler ICE space Y . Let H ⊂ π 1 (X) be a finitely generate subgroup and X H → X the corresponding covering. Proposition 3.4 will enable us, under suitable hypotheses, to pass from a core of X to a finite-sheeted pre-cover X. The next step is to add vertex spaces so that we can apply Stallings' Principle. Furthermore, we would like to do this in such a way that the resulting cover retracts onto the pre-coverX. To this end, we will need to construct finite-sheeted covers of the vertex spaces with special properties. The next lemma does this for T . Lemma 3.5 Let T be a torus and δ : S 1 → T an essential loop. Then for every positive integer d there exists a finite-sheeted coveringT d → T so that δ has a single elevationδ toT d , andδ is of degree d.
Proof. Since π 1 (T ) is abelian we can assume that δ is a based loop. Let π 1 (S 1 ) = t and consider the decomposition where C is maximal cyclic with δ * t ∈ C and A is free abelian. Setting π 1 (T d ) = dδ * t ⊕ A gives the required covering.
Note that π 1 (T d ) retracts onto δ = dδ * t . For Y , the inductive hypothesis that π 1 (Y ) admits local retractions will ensure the existence of the necessary covers.
Lemma 3.6 Let Y be a space such that π 1 (Y ) has local retractions and let δ : S 1 → Y be a based essential closed loop. Then for every positive integer d there exists a finite-sheeted coveringŶ d → Y so that δ has an elevationδ toŶ d of degree d and π 1 (Ŷ d ) retracts onto δ .
If we don't require π 1 (Ŷ d ) to retract onto δ then Lemma 3.6 can be significantly strengthened for residually free groups. Indeed, whenever the fundamental group of a space Y is residually torsion-free nilpotent, the cover Y d can be taken to be normal, so every elevation of δ toŶ d is of degree d (see Remark 3.1 of [6]). By a theorem of W. Magnus, free groups (and hence residually free groups) are residually torsion-free nilpotent. However, we will not need this strengthening. Proposition 3.7 (Completing a finite-sheeted pre-cover to a cover) Let X be an ICE space constructed by gluing together a torus T and a simpler ICE space Y , as above. Assume that π 1 (Y ) admits local retractions. LetX → X be a finite-sheeted pre-covering. Then there exists an inclusion X →X extendingX → X to a coveringX → X such that π 1 (X) retracts onto π 1 (X).
Proof. Let d be a positive integer. By Lemma 3.6 there exists a finitesheeted coveringȲ d → Y with an elevation of ∂ + of degree d. Likewise, by Lemma 3.5 there exists a finite-sheeted coveringT d → T with just one elevation of ∂ − , and that elevation is of degree d.
Let m d be the number of hanging elevations of ∂ − toX of degree d. Consider the pre-coverX Let p d be the number of degree d elevations of ∂ + toX + and q d the number of degree d elevations of ∂ − toX + . By construction, n d = p d − q d ≥ 0. Pairing up every degree d hanging elevation of ∂ − inX with a hanging elevation of ∂ + in someȲ d , we can assume thatX + is connected and has n d hanging elevations of ∂ + of degree d. Consider the pre-cover By construction, there exist n d degree d hanging elevations of ∂ + toX 0 and n d degree d elevations of ∂ − toX 0 . So, by Stallings' Principle,X 0 extends to a finite-sheeted coverX of X.
Each π 1 (T d ) retracts onto the edge group to which it was glued; then each π 1 (Ȳ d ) retracts onto the edge group to which it was glued. So π 1 (X) retracts onto π 1 (X).

The induction
In this section we prove our main theorem.
Theorem 3.8 Let X be an ICE space, let H ⊂ π 1 (X) be a finitely generated subgroup and let X H → X be the corresponding covering. Suppose L is a (possibly empty) finite, independent set of loops that generate maximal abelian subgroups of π 1 (X). Then X H is tame over L.
The proof of Theorem 3.8 is an induction. The inductive step is similar to the proof of Corollary 1.8, which provides the base step.
Proof of Theorem 3.8. Fix a finitely generated subgroup H ⊂ π 1 (X) and let X H → X be the corresponding covering space. For notational convenience, divide L into hyperbolic loops {δ i : D i → X} and elliptic loops { i : E i → X}. After modifying by a homotopy, we can assume that each i is contained in a vertex of space of X. Let ∆ ⊂ X H be a finite subcomplex, and as in the definition of tameness let {δ H j : D H j → X H } be a finite collection of infinite-degree elevations of the {δ i } and { j : E j → X H } a finite collection of infinite-degree elevations of the { i }.
Since H is finitely generated, there exists a core X ⊂ X H . Enlarging X if necessary, we can assume that ∆ ⊂ X , that each δ H j restricts to an elevation δ j : D j → X and that the image of each j is contained in X .
By Lemma 2.24 we can enlarge X further and assume that the δ j are disparate. For all sufficiently large d, since D j is a finite collection of compact intervals, there exists an embedding into a circle D j →D j so that the dsheeted coveringD j → D i extends D j → D i ; so by Lemma 2.23 we can extend X to a pre-coverX such that each δ j extends to a full elevation δ j :D j →X.
Enlarging ∆ again, we can assume that the images of theδ j are contained in ∆. By Proposition 3.4 there exists some intermediate finite-sheeted pre-coveringX →X → X into which ∆ injects, and such that each j descends to a full elevation j :Ê j →X withÊ j → E i a covering of degree d. Since ∆ injects intoX we already have thatδ j descends to an elevation δ j :D j →X.

Conclusion
Theorems A and B follow easily. By Lemma 3.2 and Theorem 3.8, groups in ICE are subgroup separable and have local retractions. Applying Theorem 1.15 and the facts that subgroup separability and local retractions pass to subgroups, it follows that limit groups enjoy both these properties.  It follows from Corollary 3.9 that limit groups have solvable generalized word problem. Corollary 3.10 and Lemma 1.5 together imply that finitely generated subgroups of limit groups are quasi-isometrically embedded.
Corollary 3.12 Every finitely generated subgroup of a limit group is quasiisometrically embedded.
F. Dahmani proved in [5] the stronger result that finitely generated subgroups of limit groups are quasi-convex in the sense of convergence groups. In contrast, our techniques are purely topological, and indeed the notion of quasi-convexity is not well-defined without introducing some geometry.
In this context one can deduce more from Corollary 3.12. Emina Alibegovic and Mladen Bestvina showed in [1] that any limit group G acts geometrically on a CAT(0) space with isolated flats X. Christopher Hruska proved that in this situation a finitely generated subgroup H of G is quasiconvex with respect to the action on X if and only if H is quasi-isometrically embedded in G (Theorem 1.1 of [9], see also Proposition 4.1.6 of [10]). Combining these results with Corollary 3.12 we obtain a quasi-convexity result.
Corollary 3.13 Let G be a limit group with an action ρ by isometries on a CAT(0) space with isolated flats. Then every finitely generated subgroup H of G is quasi-convex with respect to the action ρ.
The proof of Theorem 3.8 has deeper consequences for the separability properties of ICE groups.
Corollary 3.14 Let G ∈ ICE and let H be a vertex or edge group of G. Then G is coset separable with respect to H.
Proof. As usual, think of G as the fundamental group of an ICE space X. We will prove the corollary when H is a vertex group of G, corresponding to a vertex space V of X. The proof when H corresponds to an edge group is the same.
Let H ⊂ G be a finitely generated subgroup and let X H be the corresponding cover of X. Assume that HgH = HhH . Translating this to the language of elevations, this is equivalent to asserting that the elevations of the inclusion map V → X to the cover X H are non-isomorphic (in particular, they have disjoint images). The image of each of these elevations is precisely a vertex space of X H . Let V g be the vertex space corresponding to HgH and let V h be the vertex space corresponding to HhH .
Let X be a core for X H . Enlarging X if necessary, we can assume that V g and V h are both contained in X . The proof of Theorem 3.8 constructs an intermediate finite-sheeted coverX → X in which distinct vertex and edge spaces of X descend to distinct vertex and edge spaces ofX. In particular, Hgπ 1 (X) = Hhπ 1 (X), as required.