Limit groups and groups acting freely on ^n-trees

07/21/2003 ∙ by Vincent Guirardel, et al. ∙ University Paul Sabatier (Toulouse, France) 0

We give a simple proof of the finite presentation of Sela's limit groups by using free actions on ^n-trees. We first prove that Sela's limit groups do have a free action on an ^n-tree. We then prove that a finitely generated group having a free action on an ^n-tree can be obtained from free abelian groups and surface groups by a finite sequence of free products and amalgamations over cyclic groups. As a corollary, such a group is finitely presented, has a finite classifying space, its abelian subgroups are finitely generated and contains only finitely many conjugacy classes of non-cyclic maximal abelian subgroups.

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1 Introduction

Limit groups have been introduced by Z. Sela in the first paper of his solution of Tarski’s problem [Sel01]. These groups appeared to coincide with the long-studied class of finitely generated fully residually free groups (see [Bau67], [Bau62], [KM98a, KM98b], [Chi95] and references).

A limit group is a limit of free groups in the space of marked groups. More precisely, if is a fixed integer, a marked group is a group together with an ordered generating family . Two marked groups and are close to each other in this topology if for some large , and have exactly the same relations of length at most (see section 2.1).

Limit groups have several equivalent characterizations: a finitely generated group is a limit group if and only if it is fully residually free, if and only if it has the same universal theory as a free group, if and only if it is a subgroup of a non-standard free group ([Rem89, CG]).

One of the main results about limit groups is a structure theorem due to Kharlampovich-Myasnikov, Pfander and Sela ([KM98a, KM98b, Pfa97, Sel01]). This theorem claims that a limit group can be inductively obtained from free abelian groups and surface groups by taking free products and amalgamations over (see Th.7.1 below). This structure theorem implies that a limit group is finitely presented, and that its abelian subgroups are finitely generated. The goal of the paper is to give a simpler proof of this result in the broader context of groups acting freely on -trees.

After completing this work, the author learnt about the unpublished thesis of Shalom Gross, a student of Z. Sela, proving the finite presentation of finitely generated groups having a free action on an -tree ([Gro98]). Both proofs deeply rely on Sela’s structure theorem for super-stable actions of finitely generated groups on -trees ([Sel97, Th.3.1], see also Th.5.2 below). However, Shalom does not state a dévissage theorem over cyclic groups, but over finitely generated abelian groups.

Let’s recall briefly the definition of a -tree. Given a totally ordered abelian group , there is a natural notion of -metric space where the distance function takes its values in . If is archimedean, then is isomorphic to a subgroup of and we have a metric in the usual sense. When is not archimedean, there are elements which are infinitely small compared to other elements. A typical example is when endowed with the lexicographic ordering.

A -tree may be defined as a geodesic -hyperbolic -metric space. Roughly speaking, an -tree may be thought of as a kind of bundle over an -tree where the fibers are (infinitesimal) -trees.

In his list of research problems, Sela conjectures that a finitely generated group is a limit group if and only if it acts freely on an -tree ([Sel]). However, it is known that the fundamental group of the non-orientable surface of Euler characteristic is not a limit group since three elements in a free group satisfying must commute ([Lyn59],[Chi01, p.249]). But this group acts freely on a -tree: can be obtained by gluing together the two boundary components of a twice punctured projective plane, so can be written as an HNN extension . The -tree can be roughly described as the Bass-Serre tree of this HNN extension, but where one blows up each vertex into an infinitesimal tree corresponding to a Cayley graph of (see [Chi01, p.237] for details).

In this paper, we start by giving a proof that every limit group acts freely on an -tree. This is an adaptation a theorem by Remeslennikov saying that a fully residually free groups act freely on a -tree where has finite -rank, i. e.  is finite dimensional ([Rem92], see also [Chi01, th. 5.10]). However, Remeslennikov claims that can be chosen finitely generated, but this relies on a misquoted result about valuations (see section 3).

We actually prove that there is a closed subspace of the space of marked groups consisting of groups acting freely on -trees. In the following statement, an action of a group on a Bruhat-Tits tree is the action on the Bruhat-Tits tree of induced by a morphism where is a valuation field. Note that may vary with .

thm_closed_aux thmsectionsubsectionsubsubsection thm_closed_sauve

Theorem 1.1 (Acting freely on Bruhat-Tits trees is closed, see also [Rem92]).

Let be the set of marked groups having a free action on a Bruhat-Tits tree.

Then is closed in , and consists in groups acting freely on -trees where has dimension at most over . In particular, consists in groups acting freely on -trees where has the lexicographic ordering.

thm_closed_aux

Remeslennikov_aux thmsectionsubsectionsubsubsection Remeslennikov_sauve

Corollary 1.2 ([Rem92]).

A limit group has a free action on an -tree.

Remeslennikov_aux

Remark.

As a corollary of their study of the structure of limit groups ([KM98a, KM98b]), Kharlampovich and Myasnikov prove the more precise result that a limit group is a subgroup of an iterated free extension of centralizers of a free group, and has therefore a free action on a -tree ([KM98b, Cor.6]). An alternative proof of this fact using Sela’s techniques is given in [CG].

The main result of the paper is the following structure theorem for groups acting freely on -trees (see theorem 7.2 for a more detailed version). In view of the previous corollary, this theorem applies to limit groups.

devissage_simple_aux thmsectionsubsectionsubsubsection devissage_simple_sauve

Theorem 1.3 (Dévissage theorem, simple version. See also [Gro98, Cor. 6.6]).

Consider a finitely generated, freely indecomposable group having a free action on an -tree. Then can be written as the fundamental group of a finite graph of groups with cyclic edge groups and where each vertex group is finitely generated and has a free action on an -tree.

devissage_simple_aux

For , Rips theorem says that (which is supposed to be freely indecomposable) is either a free abelian group, or a surface group (see [GLP94, BF95]). Hence, a limit group can be obtained from abelian and surface groups by a finite sequence of free products and amalgamations over . It is therefore easy to deduce the following result:

cor_FP_aux thmsectionsubsectionsubsubsection cor_FP_sauve

Corollary 1.4 (See also [Gro98]).

Let be a finitely generated group having a free action on an -tree. Then

is finitely presented ([Gro98, Cor.6.6]);

if is not cyclic, then its first Betti number is at least ;

there are finitely many conjugacy classes of non-cyclic maximal abelian subgroups in , and abelian subgroups of are finitely generated. More precisely, one has the following bound on the ranks of maximal abelian subgroups:

where the sum is taken over the set of conjugacy classes of non-cyclic maximal abelian subgroups of , and where denotes the first Betti number of ;

has a finite classifying space, and the cohomological dimension of is at most where is the maximal rank of an abelian subgroup of .

cor_FP_aux

Remark.

A combination theorem by Dahmani also shows that is hyperbolic relative to its non-cyclic abelian subgroups ([Dah02]).

Corollary ([Sel01, KM98a, KM98b, Pfa97]).

A limit group is finitely presented, its abelian subgroups are finitely generated, it has only finitely many conjugacy classes of maximal non-cyclic abelian subgroups, and it has a finite classifying space.

Finally, we can also easily derive from the dévissage theorem the existence of a principal splitting, a major step in Sela’s proof of the finite presentation of limit groups (see corollary 7.4 and [Sel01, Th.3.2]).

Unlike Sela’s proof, the proof we give doesn’t need any JSJ theory, and does not use the shortening argument. The proof is also much shorter than the one by Kharlampovich-Myasnikov in [KM98a, KM98b] using algebraic geometry over groups, and the study of equations in free groups.

The paper is organized as follows: after some premilinaries in section 2, section 3 is devoted to the proof of the fact that limit groups act freely on -trees. Section 4 sets up some preliminary work on graph of actions on -trees, which encode how to glue equivariantly some -trees to get a new -tree. In section 5, starting with a free action of a group on an -tree , we study the action on the -tree obtained by identifying points at infinitesimal distance, and we deduce a weaker version of the dévissage Theorem where we obtain a graph of groups over (maybe non-finitely generated) abelian groups. Section 6 contains the core of the argument: starting with a free action of on an -tree , we build a free action on an -tree such that the -tree has cyclic arc stabilizers. The dévissage theorem and its corollaries will then follow immediately, as shown in section 7.

2 Preliminaries

2.1 Marked groups and limit groups

Sela introduced limit groups in [Sel01]. For background about Sela’s limit groups, see also [CG] or [Pau03].

A marked group is a finitely generated group together with a finite ordered generating family . Note that repetitions may occur in , and some generators may be the trivial element of . Consider two groups and together with some markings of the same cardinality and . A morphism of marked groups is a homomorphism sending on for all . Note that there is at most one morphism between two marked groups, and that all morphisms are epimorphisms.

A relation in is an element of the kernel of the natural morphism sending to where is the free group with basis . Note that two marked group are isomorphic if and only if they have the same set of relations.

Given any fixed , we define to be the set of isomorphism classes of marked groups. It is naturally endowed with the topology such that the sets defined below form a neighbourhood basis of . For each and each , is the set of marked groups such that and have exactly the same relations of length at most . For this topology, is a Hausdorff, compact, totally disconnected space.

Definition 2.1.

A limit group is a marked group which is a limit of markings of free groups in .

Actually, being a limit group does not depend on the choice of the generating set. Moreover, limit groups have several equivalent characterizations: a finitely generated group is a limit group if and only if it is fully residually free, if and only if it has the same universal theory as a free group, if and only if it is a subgroup of a non-standard free group ([Rem89, CG]). We won’t need those characterizations in this paper.

2.2 -trees

For background on -trees, see [Bas91, Chi01].

Totally ordered abelian groups

A totally ordered abelian group is an abelian group with a total ordering such that for all , . Our favorite example will be , with the lexicographic ordering. In all this paper, will always be endowed with its lexicographic ordering. To fix notations, we use the little endian convention: the leftmost factor will have the greatest weight. More precisely, if and are totally ordered abelian groups, the lexicographic ordering on is defined by if or ( and ).

A morphism between two totally-ordered abelian groups is a non-decreasing group morphism. Given , the subset is called the segment between and . A subset is convex if for all , . The kernel of a morphism is a convex subgroup, and if is a convex subgroup, then has a natural structure of totally ordered abelian group. By proper convex subgroup of , we mean a convex subgroup strictly contained in .

The set of convex subgroups of is totally ordered by inclusion. The height of is the (maybe infinite) number of proper convex subgroups of . Thus, the height of is . is archimedean if its height is at most 1. It is well known that a totally ordered abelian group is archimedean if and only if it is isomorphic to a subgroup of (see for instance [Chi01, Th.1.1.2])

If is a convex subgroup, then any element may be thought as infinitely small compared to an element since for all , . Therefore, we will say that an element in is infinitesimal if it lies in the maximal proper convex subgroup of , which we casually denote by . Similarly, for , we will identify with the corresponding convex subgroup of . The magnitude of an element is the smallest such that . Thus is infinitesimal if and only if its magnitude is at most .

Given a totally ordered abelian group , has a natural structure of a totally ordered abelian group by letting if and only if .

-metric spaces and -trees

A -metric space is a set endowed with a map satisfying the three usual axioms of a metric: separation, symmetry and triangle inequality. The set itself is a -metric space for the metric . A geodesic segment in is an isometric map from a segment to a subset of . A -metric space is geodesic if any two points are joined by a geodesic segment. We will denote by a geodesic segment between two points in (which, in this generality, might be non-unique).

Note that even in a set like , the upper bound is not always defined so one cannot easily define a -valued diameter (see however [Chi01, p.99] for a notion of diameter as a interval in ). Nevertheless, we will say that a subset of a -metric space is infinitesimal if the distance between any two points of is infinitesimal. Similarly, we define the magnitude of as the smallest such that the distance between any two points of has magnitude at most .

We give two equivalent definitions of a -tree. The equivalence is proved for instance in [Chi01, lem. 2.4.3, p. 71].

Definition 2.2.

A -tree is a geodesic -metric space such that

is -hyperbolic in the following sense:

Equivalently, a geodesic -metric space is a -tree if

the intersection of any two geodesic segments sharing a common endpoint is a geodesic segment

if two geodesic segments intersect in a single point, then their union is a geodesic segment.

Remark.

In the first definition, the second condition is automatic if , which is the case for .

It follows from the definition that there is a unique geodesic joining a given pair of points in a -tree.

Clearly, itself is -tree. Another simple example of a -tree is the vertex set of a simplicial tree : endowed with the combinatorial distance is a -tree.

2.3 Killing infinitesimals and extension of scalars

The following two operations are usually known as the base change functor.

Killing infinitesimals.

Consider a convex subgroup (a set of infinitesimals), and let . If , we will usually take , so that . Consider a -metric space . Then the relation defined by is an equivalence relation on , and the -metric on provides a natural -metric on . We say that is obtained from by killing infinitesimals. Clearly, if is a -tree, then is a -tree. Thus, killing infinitesimals in an -tree provides an -tree . By extension, we will often denote -trees with a bar.

Extension of scalars

Consider a -tree , and an embedding (for example, one may think of ). Then may be viewed as a -metric space, but it is not a -tree if is not convex in : as a matter of fact, is not geodesic as a -metric space (there are holes in the geodesics). However, there is a natural way to fill the holes:

Proposition 2.3 (Extension of scalars, see [Chi01, Th. 4.7,p. 75]).

There exists a -tree and an isometric embedding which is canonical in the following sense: if is another -tree with an isometric embedding , then there is a unique -isometric embedding commuting with the embeddings of in and .

For example, take to be the -tree corresponding corresponding to the set of vertices of a simplicial tree . Then the embedding gives an -tree which is isometric to the geometric realization of .

Remark.

The proposition also holds if one only assumes that is -hyperbolic. In this case, taking , one gets a natural -tree containing .

2.4 Subtrees

A subtree of a -tree is a convex subset of , i. e. such that for all , . A subtree is non-degenerate if it contains at least two points. One could think of endowing , and , with the order topology. However, this is usually not adapted. For instance: is not connected with respect to this topology for . This is why we need a special definition of a closed subtree. The definition coincides with the topological definition for -trees.

Definition 2.4 (closed subtree).

A subtree is a closed subtree if the intersection of with a segment of is either empty or a segment of .

There is a natural projection on a closed subtree. Consider a base point . Then for any point , there is a unique point such that . One easily checks that does not depend on the choice of the base point ( is the bridge between and , see [Chi01]). The point is called the projection of on .

Remark.

The existence of a projection is actually equivalent to the fact that the subtree is closed. Be aware that a non-trivial proper convex subgroup of is never closed in . In particular, the intersection of infinitely many closed subtrees may fail to be closed.

A linear subtree of is a subtree in which any three points are contained in a segment. It is an easy exercise to prove that a maximal linear subtree of is closed in . Finally, any linear subtree is isometric to a convex subset of and any two isometries differ by an isometry of .

2.5 Isometries

An isometry of a -tree can be of one of the following exclusive types:

elliptic: has a fix point in

inversion: has no fix point, but does

hyperbolic: otherwise. In all cases, the set is called the characteristic set of .

If is elliptic, is the set of fix points of which is a closed subtree of . Moreover, for all , the midpoint of exists and lies in .

If is an inversion, then . Actually, for any , so has no midpoint in . In particular, if (which occurs for instance if ), inversions don’t exist. Moreover, one can perform the analog of barycentric subdivision for simplicial trees to get rid of inversions: consider , and let be the -tree obtained by the extension of scalars . Then the natural extension of to fixes a unique point in (in particular, is elliptic in ). If is elliptic or is an inversion, its translation length is defined to be .

If is hyperbolic, then the set is non-empty, and is a maximal linear subtree of , and is thus closed in . It is called the axis of . Moreover, the restriction of to is conjugate to the action of a translation on a -invariant convex subset of for some positive . The translation length of is the element . If is the projection of on , then for , .

Note that it may happen that is not isometric to . For instance, if , the axis of an element with infinitesimal translation length can be of the form where is any non-empty interval in which can be open, semi-open or closed.

If is hyperbolic, then for all , the projection of on is the projection of on . In particular, if the midpoint of exists, then it lies in . It also follows that if is hyperbolic and if are aligned (in any order) then they lie on the axis of .

If an abelian group acts by isometries on -tree and contains a hyperbolic element , then all the hyperbolic elements of have the same axis , contains no inversion, and all elliptic elements fix . We say that is the axis of the abelian group . The axis of can be characterized as the only closed -invariant linear subtree of , or as the only maximal -invariant linear subtree of .

2.6 Elementary properties of groups acting freely on -trees

We now recall some elementary properties of groups acting freely (without inversion) on -trees. They are proved for instance in [Chi01].

Lemma 2.5.

Let be a group acting freely without inversion on a -tree. Then

is torsion free;

two elements commute if and only if they have the same axis. If they don’t commute, the intersection of their axes is either empty or a segment ([Chi01], proof of lem. 5.1.2 p.218 and Rk p.111)

maximal abelian subgroups of are malnormal (property CSA) and is commutative transitive: the relation of commutation on is transitive ([Chi01, lem. 5.1.2 p.218])

Remark.

Property CSA implies that is commutative transitive.

A result known as Harrison Theorem, proved by Harrison for -trees and by Chiswell and Urbanski-Zamboni for general -trees, says that a -generated group acting freely without inversion on a -tree is either a free group or a free abelian group. (see [Chi94, UZ93, Har72]). We won’t use this result in this paper.

3 A limit group acts freely on an -tree

The goal of this section is to prove that limit groups act freely on -trees. This is an adaptation of an argument by Remeslennikov concerning fully residually free groups ([Rem92], see also [Chi01, th. 5.5.10 p. 246]). Note that it is claimed in [Rem92] that finitely generated fully residually free groups act freely on a -tree where is a finitely generated ordered abelian group. However, the proof is not completely correct since it relies on a misquoted result about valuations (Th.3 in [Rem92]) to which there are known counterexamples (for any subgroup , there is valuation on , extending the trivial valuation on , whose value group is [ZS75, ch.VI,§15, ex.3,4] or [Kuh, Th 1.1]). Nevertheless, Remeslennikov’s argument proves the following weaker statement: a finitely generated fully residually free group acts freely on a -tree where has finite -rank, i. e.  is finite dimensional over .

The fact that a limit group acts freely on an tree will be deduced from a more general result about group acting freely on Bruhat-Tits trees. But we first state a simpler result in this spirit (see also [GS94, GS93]). Remember that denotes the space of groups marked by a generating family of cardinality .

Proposition 3.1 (Acting freely on -trees is closed).

Let be the set of marked groups having a free action without inversion on some -tree ( may vary with the group).

Then is closed in .

We won’t give the proof of this result since this proposition is not sufficient for us as it does not give any control over . This is why we rather prove the following more technical result.111The proof is actually very similar to the proof of the more technical result: instead of taking ultraproducts of valuated fields, take an ultraproduct of trees to get a free action without inversion on a -tree (see also [Chi01, p.239] where the behavior -trees under ultrapowers is studied in terms of Lyndon length functions).

For general information of the action of on its Bruhat-Tits -tree where a field, and is a valuation, see for instance [Chi01, §4.3,p.144]. Essentially, we will only use the existence of the Bruhat-Tits -tree and the formula for the translation length of a matrix : . Also note that the action of on its Bruhat-Tits tree has no inversion (however, there may be inversions in ).

Definition 3.2 (Action on a Bruhat-Tits tree.).

By an action of on a Bruhat-Tits tree, we mean an action of on the Bruhat-Tits -tree for induced by a morphism where is a valuated field with values in .

thm_closed_sauve thmsectionsubsectionsubsubsection

Theorem 3.3 (Acting freely on Bruhat-Tits trees is closed, see also [Rem92]).

Let be the set of marked groups having a free action on a Bruhat-Tits tree.

Then is closed in , and consists in groups acting freely on -trees where has dimension at most over . In particular, consists in groups acting freely on -trees where has the lexicographic ordering.

Remeslennikov_sauve thmsectionsubsectionsubsubsection

Corollary 3.4 ([Rem92]).

A limit group has a free action on an -tree.

Proof of the corollary.

This follows from the theorem above since a free group acts freely on a Bruhat-Tits tree. ∎

Proof of the Theorem.

We first prove that is closed. Let be a sequence of marked groups converging to . For each index , consider a field and a valuation and an embedding such that acts freely without inversions on the corresponding Bruhat-Tits tree .

Consider an ultrafilter on , i. e. a finitely additive measure of total mass (a mean), defined on all subsets of , and with values in , and assume that this ultrafilter is non-principal, i. e. that the mass of finite subsets is zero. Say that a property depending on is true -almost everywhere if . Note that a property which is not true almost everywhere is false almost everywhere. Given a sequence of sets , the ultraproduct of is the quotient where is the natural equivalence relation on defined by equality -almost everywhere.

Consider the ultraproduct of the fields , the ultraproduct of the groups , and the ultraproduct of the totally ordered abelian groups . As a warmup, we prove the easy fact that the natural ring structure on makes it a field: if in , then for almost all , , and is defined for almost every index , and defines an inverse for in .

Similarly, is a group, and a totally ordered abelian group (for the total order if and only if almost everywhere). Now consider the map defined by , and the map defined by . Then is a valuation on , and a monomorphism of groups. We denote by the Bruhat-Tits tree of .(222It may also be checked that is actually the ultraproduct of the -trees .)

Now, given a field with a valuation , a subgroup acts freely without inversions on the corresponding Bruhat-Tits tree if and only if the translation length of any element is non-zero. But the translation length of a matrix can be computed in terms of the valuation of its trace by the formula , so the freeness (without inversion) of the action translates into for all ([Chi01, lem.4.3.5 p.148]). Therefore, since for all and all , has negative valuation, all the elements satisfy , which means that acts freely without inversion on .

Finally, there remains to check that the marked group embeds into (see for instance [CG]). We use the notation and . Consider the family of elements of defined by . The definition of the convergence of marked groups says that if an -word represents the trivial element (resp. a non-trivial element) in , then for sufficiently large, the corresponding -word is trivial (resp. non-trivial) in . Since is non-principal, this implies that the corresponding -word is trivial (resp. non-trivial). This means that the map sending to extends to an isomorphism between and . Therefore, , so is closed.

We now prove the fact that any group in acts freely on a -tree where has dimension at most . So consider an embedding where has a valuation such that the induced action of on the Bruhat-Tits tree for is free without inversion. Consider the subfield generated by the coefficient of the matrices . Since the matrices have determinant , can be written as where is the prime subfield of . Let be the value group of . Since embeds in , acts freely on the corresponding Bruhat-Tits -tree. We now quote a result about valuations which implies that has finite -rank.

Theorem 3.5 ([Bou64, cor 1 in VI.10.3]).

Let be a finitely generated extension of , and a valuation. Denote by (resp. ) the corresponding value group. Then the

-vector space

has dimension at most .

Taking , one gets that has -rank at most since is either trivial or isomorphic to .

Using the extension of scalars (base change functor), there remains to prove that if a totally ordered group has finite -rank, then it is isomorphic to a subgroup of . ∎

Lemma 3.6.

Consider a totally ordered of -rank . Then is isomorphic (as an ordered group) to subgroup of with its lexicographic ordering.

Remark.

However, is usually not isomorphic to with its lexicographic ordering as shows an embedding of into .

Proof.

We first check that the height of is at most (see [Bou64, prop 3 in VI.10.2]). First, embeds into , so we may replace by and assume that is a totally ordered -vector space of dimension . Any convex subgroup is a vector subspace in since if , for all , since . Now the height of is at most since a chain of convex subgroups is a chain of vector subspaces.

We now prove by induction that a totally ordered group of finite height embeds as an ordered subgroup of with its lexicographic ordering. Once again, one can replace by without loss of generality. We argue by induction on the height. If has height , i. e. if is archimedean, then embeds in (see for instance [Chi01, Th.1.1.2]). Now consider the maximal proper convex subgroup of , and let . Since and are vector spaces, one has algebraically .

The fact that is convex in implies that the ordering on corresponds to the lexicographic ordering on ([Bou64, lemma 2 in VI.10.2]). Indeed, one first easily checks that any section is increasing. Now let’s prove that the isomorphism defined by is increasing for the lexicographic ordering on . So assume that . If , then . If , then since otherwise, one would have , hence by convexity, a contradiction.

Finally, by induction hypothesis, embeds as an ordered subgroup of and embeds as an ordered subgroup of , so embeds as an ordered subgroup of . ∎

4 Gluing -trees

The goal of this section is to define graph of actions on -trees which show how to glue actions on -trees along closed subtrees to get another action on a -tree, and to give a criterion for the resulting action to be free. We will finally study more specifically gluings of -trees along points, and show that a decomposition of an -tree into a graph of actions on -trees above points correspond to a transverse covering of by closed subtrees.

4.1 Gluing -trees along points

Here, we recall that one can glue trees together along a point to get a new -tree (see [Chi01, Lem. 2.1.13]).

Lemma 4.1 ([Chi01, Lemma 2.1.13]).

Let be a -tree, and be a family of -trees. Assume that and that for all , . Let and let defined by: ; ; for ; for .

Then is a -tree.

4.2 Gluing two trees along a closed subtree

The following gluing construction will be used for gluing trees along maximal linear subtrees.

Assume that we are given two -trees , two closed subtrees and , and an isometric map . By definition of a closed subtree, we have two orthogonal projections for .

Let , and let be the equivalence relation on generated by for all . The set is now endowed with the following metric which extends on : if and , we set

(2)

To prove the last equality, introduce ; then for any ,

Lemma 4.2.

With the definitions above, is a -tree. Moreover, any closed subtree of is closed in .

Proof.

Let . Then can be viewed as the tree on which are glued some subtrees of at some points. More precisely, for , let , and similarly, for , let . Then, because of the formula (4.2) for the metric, is isometric to the -tree obtained by gluing the trees and on along the point as in lemma 4.1. Therefore, by lemma 4.1, is an -tree.

Now let be a closed subtree. Let’s prove that is closed in . Consider , , and let’s prove that there exists such that . If , then one can take to be the projection of on by hypothesis. If , let be the projection of on , and be the projection of on . Of course, . Now does not meet since it is contained in , and neither does . Thus and is closed in . ∎

4.3 Equivariant gluing: graphs of actions on -trees.

The combinatorics of the gluing will be given by a simplicial tree , endowed with an action without inversion of a group . We denote by and the set of vertices and (oriented) edges of , by and the origin and terminus of an (oriented) edge , and by the edge with opposite orientation as .

A graph of actions on trees is usually defined as a graph of groups with some additional data like vertex trees. Here, we rather use an equivariant definition at the level of the Bass-Serre tree.

Definition 4.3 (Graph of actions on -trees.).

Given a group , a -equivariant graph of actions on -trees is a triple where

is a simplicial tree,

for each vertex , is a -tree (called vertex tree),

for each edge , is an isometry between closed subtrees and such that . We call the subtrees the edge subtrees.

This data is assumed to be -equivariant in the following sense:

acts on without inversion,

acts on so that the restriction of each element of to a vertex tree is an isometry,

the natural projection (sending a point in to ) is equivariant

the family of gluing maps is equivariant: for all , , and .

The -tree dual to a graph of actions

Given a -equivariant graph of actions on -trees, we consider the smallest equivalence relation on such that for all edge and , . The -tree dual to is the quotient space . To define the metric on , one can alternatively say that is obtained by gluing successively the vertex trees along the edge trees according to lemma 4.2 in previous section. Formula (2) in previous section shows that the metric does not depend on the order in which the gluing are performed. Indeed, an induction shows that the distance between and can be computed as follows: let the edges of the path from to in , and the corresponding vertices then

where the minimum is taken over all . By finitely many applications of lemma 4.2, one gets that the gluings corresponding to finite subtrees of are -trees. Now apply the fact that an increasing union of -trees is a -tree to get that is a -tree (see [Chi01, Lem. 2.1.14]). We thus get the following lemma:

Definition 4.4 (Tree dual to a graph of actions on -trees.).

Consider a -equivariant graph of actions on -trees. The dual tree is the set endowed with the metric defined above. It is a -tree on which acts by isometries.

We say that a -tree splits as a graph of actions if there is an equivariant isometry between and .

Remark.

Consider an increasing union of trees such that is a closed subtree of each . Then is closed in . Therefore, using lemma 4.2, one gets that a closed subtree of a vertex tree is closed in . In particular, vertex trees themselves are closed in .

4.4 Gluing free actions into free actions

We next give a general criterion saying that an action obtained by gluing is free. It is stated in terms of the equivalence relation on defined above. Each equivalence class has a natural structure of a connected graph: elements of the equivalence class are vertices, put an oriented edge between two vertices and if for some edge . This graph embeds into via the map , so this graph is a simplicial tree.

Lemma 4.5 (Criterion for a graph of free actions to be free).

Consider a -equivariant graph of action on -trees. For each vertex , denote by its stabilizer, and assume that the action of on is free. Assume furthermore that each equivalence class of has finite diameter.

Then the action of on