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APPEARED IN BULLETIN OF THE

arXiv:math/9201265v1 [math.GR] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 87-112Λ-TREES AND THEIR APPLICATIONSJohn W. MorganTo most mathematicians and computer scientists the word “tree” conjures up,in addition to the usual image, the image of a connected graph with no circuits. Weshall deal with various aspects and generalizations of these mathematical trees.

(AsPeter Shalen has pointed out, there will be leaves and foliations in this discussion,but they do not belong to the trees!) In the last few years various types of trees havebeen the subject of much investigation.

But this activity has not been exposed muchto the wider mathematical community. To me the subject is very appealing for itmixes very na¨ıve geometric considerations with the very sophisticated geometricand algebraic structures.

In fact, part of the drama of the subject is guessing whattype of techniques will be appropriate for a given investigation: Will it be directand simple notions related to schematic drawings of trees or will it be notions fromthe deepest parts of algebraic group theory, ergodic theory, or commutative algebrawhich must be brought to bear? Part of the beauty of the subject is that the na¨ıvetree considerations have an impact on these more sophisticated topics.

In addition,trees form a bridge between these disparate subjects.Before taking up the more exotic notions of trees, let us begin with the graph-theoretic notion of a tree. A graph has vertices and edges with each edge havingtwo endpoints each of which is a vertex.

A graph is a simplicial tree if it containsno loops. In §1 we shall discuss simplicial trees and their automorphism groups.This study is closely related to combinatorial group theory.The later sectionsof this article shall focus on generalizations of the notion of a simplicial tree.

Asimplicial tree is properly understood to be a Z-tree.For each ordered abeliangroup Λ, there is an analogously defined object, called a Λ-tree. A very importantspecial case is when Λ = R. When Λ is not discrete, the automorphisms groupof a Λ-tree is no longer combinatorial in nature.

One finds mixing (i.e., ergodic)phenomena occurring, and the study of the automorphisms is much richer and lesswell understood. We shall outline this more general theory and draw parallels andcontrasts with the simplicial case.While the study of trees and their automorphism groups is appealing per se,interest in them has been mainly generated by considerations outside the subject.There are several basic properties of trees that account for these connections.

Aswe go along we shall explain these notions and their connections in more detail butlet me begin with an overview.1991 Mathematics Subject Classification. Primary 05C25, 54F62, 54H12, 20G99, 20F32.Received by the editors March 13, 1991This paper was given as a Progress in Mathematics Lecture at the August 8–11, 1990 meetingin Columbus, Ohioc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2J. W. MORGAN1.One-dimensionality of trees.

Trees are clearly of dimension one. This basicproperty is reflected in relations of trees to both algebraic and geometric objects.

(a) Algebraic aspects. The resurgence of the study of trees began with Serre’s book[20] on SL2 and trees.

In this book, Serre showed how there is naturally a treeassociated to SL2(K) where K is a field with a given discrete valuation. The groupSL2(K) acts on this tree.

More generally, if G(K) is a semisimple algebraic groupof real rank 1 (like SO(n, 1)) and if K is a field with a valuation with formally realresidue field, then there is a tree on which G(K) acts. The reason for this can besummarized by saying that the local Bruhat-Tits building [2] for a rank 1 groupis a tree.

The fact that the group is rank 1 is reflected in the fact that its localbuilding is one-dimensional. (One does not need to understand to entire Bruhat-Tits machinery to appreciate this connection.

In fact, one can view this case as thesimplest, yet representative, case of the Bruhat-Tits theory. )(b) Codimension-1 dual objects.

Suppose that M is a manifold whose fundamentalgroup acts on a tree. Since the tree is 1-dimensional there is a dual object in themanifold which is codimension-1.This object turns out to be a codimension-1lamination with a transverse measure.

These dynamic objects are closed related toisometry groups of trees. In fact, dynamic results for these laminations can be usedto study group actions on trees.

They also give rise to many interesting examplesof such actions.2.Negative curvature of trees. Suppose that we have a simply connectedRiemannian manifold with distance function d of strictly negative curvature.

In Mthe following geometric property holds: If A1, A2, B1, B2 are points with d(A1, A2)and d(B1, B2) of reasonable size but d(A1, B1) extremely large, the geodesics γ1and γ2 joining A1 to B1 and A2 to B2 are close together over most of their length. (See Figure 1.) The estimate on how close will depend on an upper bound for thecurvature and will go to 0 as this bound goes to −∞.

In fact, Gromov [8] hasdefined a class of negatively curved metric spaces in terms of this 4-point property.From this point of view R-trees are the most negative curved of all spaces, havingcurvature −∞, since in a tree the geodesics γ1 and γ2 coincide over most of theirlengths.We state this fact in another way. Suppose that we have a sequence on strictlynegatively curved, simply connected manifolds with curvature upper bounds going−∞, then a subsequence of these manifolds converges in a geometric sense to anR-tree.

If the manifolds in question are the universal coverings of manifolds with agiven fundamental group G, the actions of G of the manifolds in the subsequenceconverge to an action of G on the limit tree.The space of negatively curvedmanifolds with fundamental group G is completed by adding ideal points at infinitywhich are represented by actions of G on R-trees. Thisis of particular importance for the manifolds of constant negative curvature—thehyperbolic manifolds.

This of course brings us full circle since the group of auto-morphisms of hyperbolic n-space is the real rank-1 group SO(n, 1).The paper is organized in the following manner. The first section is devotedto discussing the now classical case of simplicial trees and their relationship tocombinatorial group theory and to SL2 over a field with a discrete valuation.

Herewe follow [20] closely. This material is used to motivate all other cases.Section 2 begins with the definition of a Λ-tree for a general ordered abeliangroup Λ.

We show how if K is a field with a nondiscrete valuation with value

Λ-TREES AND THEIR APPLICATIONS3Figure 1group Λ, then there is an action of SL2(K) on a Λ-tree. More generally, for anysemisimple algebraic group of real rank 1 there is a similar result.

We explain thecase of SO(n, 1) in some detail. The last part of the section discusses some of thebasics of the actions of groups on Λ trees.

We introduce the hyperbolic length ofa single isometry and define and study some of the basic properties of the space ofall projective classes of nontrivial actions. At the end of the section we discuss theconcept of base change.Section 3 discusses the application of this material to compactify the space ofconjugacy classes of representations of a given finitely presented group into a rank-1group such as SL2 or SO(n, 1).

We approach this both algebraically and geomet-rically.The main idea is that a sequence of representations of a fixed finitelygenerated group G into SO(n, 1) has a subsequence which converges modulo con-jugation either to a representation into SO(n, 1) or to an action of G on an R-tree.In the case when the representations are converging to an action on a tree, thisresult can be interpreted as saying that as one rescales hyperbolic space by factorsgoing to zero, the actions of G on hyperbolic space coming from the sequence ofrepresentations of G into SO(n, 1) converge to an action of G on an R-tree. Thiscan then be used to compactify the space of conjugacy classes of representations ofG in SO(n, 1) with the ideal points being represented as actions of G on R-trees.Section 4 we consider the relationship between R-trees and codimension-1 mea-sured laminations.

We define a codimension-1 measured lamination in a manifold.We show how ‘most’ measured laminations in a manifold M give rise to actions ofπ1(M) on R-trees. We use these to give examples of actions of groups on R-trees,for example, actions of surface groups.

We also show how any action of π1(M) onan R-tree can be dominated by a measured lamination.In the last section we give applications of the results from the previous sections tostudy the space of hyperbolic structures (manifolds of constant negative curvature)of dimension n with a given fundamental group G. We have a compactificationof this space where the ideal points at infinity are certain types of actions of Gon R-trees. We derive some consequences—conditions under which the space ofhyperbolic structures is compact.

The theme running through this section is thatto make the best use of the results of the previous sections one needs to understandthe combinatorial group-theoretic information that can be derived from an actionof a group on an R-tree. For simplicial actions this is completely understood, aswe indicate in §1.

For more general actions there are some partial results (comingmainly by using measured laminations), but no general picture. We end the articlewith some representative questions about groups acting on R-trees and give thecurrent state of knowledge on these questions.

4J. W. MORGANFigure 2For other introductions to the subject on group actions on Λ-trees see [21, 12,3, or 1].1.

Simplicial trees, combinatorial group theory, and SL2In this discussion of the ‘classical case’ of simplicial trees, we are following closelySerre’s treatment in Trees [20]. By an abstract 1-complex we mean a set V , the setof vertices, and a set E, the set of oriented edges.

The set E has a free involutionτ : e 7→e, called reversing the orientation. There is also a map∂: E →V × V,∂e = (i(e), t(e)),which associates to each oriented edge its initial and terminal vertex satisfyingi(e) = t(e).

Closely related to an abstract 1-complex is its geometric realization,which is a topological space. It is obtained by forming the disjoint unionVa ae∈EIe,where each Ie is a copy of the closed unit interval, and taking the quotient spaceunder the following relations:(i) 0 ∈Ie is identified with i(e) ∈V .

(ii) 1 ∈Ie is identified with t(e) ∈V . (iii) s ∈Ie is identified with 1 −s in Ie.The order of a vertex v is the number of oriented edges e for which i(e) = v.A 1-complex is finite if it has finitely many edges and vertices.

In this case itsgeometric realization is a compact space. Figure 2 gives a typical example of afinite 1-complex.A nonempty, connected 1-complex is a graph; a simply connected graph is asimplicial tree.

A graph is a simplicial tree if and only if it has no loops (topolog-ical embeddings of S1). Figure 3 gives the unique (up to isomorphism) trivalentsimplicial tree.

One aspect of the negative curvature of trees is reflected in the factthat the number of vertices of distance ≤n from a given vertex is 3(2n −1) + 1, anumber which grows exponentially with n.The automorphism group of a simplicial tree T , Aut(T ), is the group of all self-homeomorphisms of T which send vertices to vertices, edges to edges, and whichare linear on each edge. This is exactly the automorphism group of the abstractcomplex (V, E, ∂, τ).

An action of a group G on T is a homomorphism from G toAut(T ). The action is said to be without inversions if for all g ∈G and all e ∈Ewe have e · g ̸= e. To restrict to actions without

Λ-TREES AND THEIR APPLICATIONS5Figure 3inversions is not a serious limitation since any action of G on T becomes anaction without inversions on the first barycentric subdivision of T (obtained bysplitting each edge of T into its two halves and adding a new vertex at the centerof each edge).Given an action of a group G on a simplicial tree T there is the quotient graphΓ. Its vertices are V/G; its oriented edges are E/G.

The geometric realization ofthis complex is naturally identified with the quotient space T/G.Graphs of groups. We are now ready to relate the theory of group actions onsimplicial trees to combinatorial group theory.

The key to understanding the natureof an action of a group on a simplicial tree is the notion of a graph of groups. Agraph of groups is the following:(i) A graph Γ,(ii) for each oriented edge or vertex a of Γ a group Ga such that if e is anoriented edge then Ge = Ge, and finally,(iii) if v = i(e) then there is given an injective homomorphism Ge ֒→Gv.This graph of groups is said to be over Γ.There is the fundamental group of a graph of groups.

A topological constructionof the fundamental group goes as follows. For each edge or vertex a of Γ choose aspace Xa whose fundamental group is Ga. We can do this so that if v is a vertexof e, then there is an embedding Xe ֒→Xv realizing the inclusion of groups.

Weform the topological space X(Γ) by beginning with the disjoint unionav∈VXva ae∈EXe × I,and (a) identifying Xe×I with Xe×I via (x, t) ≡(x, 1−t) and (b) gluing Xe×{0} toXi(e) via the given inclusion. The resulting topological space has fundamental groupwhich is the fundamental group of the graph of groups.

A purely combinatorialconstruction of this fundamental group is given in [20, pp. 41–42].Example 1.

Let Γ be a single point. Then a graph over Γ is simply a group.

Itsfundamental group is the group itself.

6J. W. MORGANExample 2.

Let Γ be a graph with two vertices and a single edge connectingthem. Then a graph of groups over Γ is the same thing as an embedding of theedge group into two vertex groups.

Its fundamental group is the free product withamalgamation of the vertex groups over the edge group.Example 3. Let Γ be a graph with one vertex and a single edge, forming a loop.Then graph of groups over Γ is a group Gv and two embeddings ϕ0 and ϕ1 ofanother group Ge into Gv.

The fundamental group of this graph of groups is thecorresponding HNN-extension. A presentation of this extension is⟨Gv, s|s−1ϕ1(g)s = ϕ0(g) for all g ∈Ge⟩.In general, the fundamental group of a graph of groups over a finite graph canbe described inductively by a finite sequence of operations as in Examples 2 and 3.The fundamental group of an infinite graph of groups is the inductive limit of thefundamental groups of the finite subgraphs of groups.Thus, it is clear that the operation of taking the fundamental group of a graph ofgroups generalizes two basic operations of combinatorial group theory—free productwith amalgamation and HNN-extension.There is a natural action of the fundamental group G of a graph of groups on asimplicial tree.

To construct this action, let X be the topological space, as describedabove, whose fundamental group is the fundamental group of the graph of groups.There is a closed subset Y = `e Xe × {1/2} ⊂X which has a collar neighborhoodin X. Let eX be the universal covering of X, and let eY ⊂eX be the preimage of Y .We define the dual tree T to eY ⊂eX.

Its vertices are the components of eX −eY .Its unoriented edges are the components of eY . The vertices of an edge given by acomponent eY0 of eY are the two components of eX −eY which have eY0 in their closure.The fact the eX is simply connected implies that T is contractible and hence is asimplicial tree.There is a natural action of G on eX.

This action leaves eY invariant and hencedefines an action of G on T . It turns out that, up to isomorphism, this actionof G on T is independent of all the choices involved in its construction.It iscalled the universal action associated with the graph of groups decomposition of G.The quotient T/G is naturally identified with the original graph.

Notice that thestabilizer of a vertex or edge a of T is identified, up to conjugation, with the groupin the graph of groups indexed by the image of a in T/G.Here, we see for the first time the duality relationship between trees and codimen-sion-1 subsets.Structure theorem for groups acting on simplicial trees. The main resultin the theory of groups actions on simplicial trees is a converse to this constructionfor a graph of groups decomposition of G. It says:Theorem 1.

Let G × T →T be an action without inversions of a group on asimplicial tree. Then there is a graph of groups over the graph T/G and an iso-morphism from the fundamental group of this graph of groups to G in such a waythat the action of G on T is identified up to isomorphism with the universal actionassociated with the graph of groups.

Λ-TREES AND THEIR APPLICATIONS7Corollary 2. Let H act on a simplicial tree T .

Suppose that no point of T is fixedby all h ∈H. The H has a HNN-decomposition or has a nontrivial decompositionas a free product with amalgamation.Proof.

At the expense of subdividing T we can suppose that H acts without in-versions.Consider the quotient graph T/H.We have a graph of groups overT/H whose fundamental group is identified with H. There is a minimal subgraphΓ ⊂T/H such that the fundamental group of the restricted graph of groups overΓ includes isomorphically into H. This graph cannot be a single vertex since theaction of H on T does not fix any point. If Γ has a separating edge e, there isa nontrivial free product with amalgamation decomposition for H as Ha and Hbamalgamated along Ge, where Ha and Hb are the fundamental groups of the graphsof groups over the two components of Γ −e.

If Γ has a nonseparating edge e′, thenthere is an HNN-decomposition for H given by the two embeddings of Ge into thefundamental group of the graph of groups over Γ −e′.□Example 4. Let M be a manifold and let N ⊂M be a proper submanifold ofcodimension 1.

Let fM be the universal covering of M and let eN ⊂fM be thepreimage of N. Then eN is collared in fM. We define a tree dual to eN ⊂fM.

Itsvertices are the components of fM −eN. Its edges are the components of eN.

Wedefine the endpoints of an edge associated to eN0 to be the components of fM −eNcontaining eN0 in their closure. Since eN is collared in fM, it follows that each edgehas two endpoints.

Thus, we have defined a graph. Since fM is simply connected,this graph is a tree.

The action of π1(M) on fM defines an action of π1(M) on T .The quotient is the graph dual to N ⊂M.Applications. As a first application notice that a group acts freely on a simplicialtree if and only if it is a free group.

If G acts freely on a tree, then G is identifiedwith the fundamental group of the graph T/G, which is a free group. Conversely,if G is free on the set W, then the wedge of circles indexed by W has fundamentalgroup identified with G. The group G acts freely on the universal covering of thiswedge, which is a tree.As our first application we have the famousCorollary 3 (Schreier’s theorem).

Every subgroup of a free group is free.Using the relation between the number of generators of a free group and theEuler characteristic of the wedge of circles, one can also establish the Schreierindex formula.If G is a free group of rank r, and if G′ ⊂G is a subgroup of index n, then G′is a free group of rank n(r −1) + 1.One can also use the theory of groups acting on trees to give a generalization ofthe Kurosh subgroup theorem. Here is the classical statement.Theorem (Kurosh subgroup theorem).

Let G = G1 ∗G2 be a free product. LetG′ ⊂G be a subgroup.

Suppose that G′ is not decomposable nontrivially as a freeproduct. Then either G′ ∼= Z or G′ is conjugate in G to a subgroup of either G1 orG2.Proof.

The decomposition G = G1 ∗G2 gives an action of G on a simplicial tree Twith trivial edge stabilizers and with each vertex stabilizer being conjugate in G toeither G1 or G2. Consider the induced action of G′ on T .

Then G′ has a graph of

8J. W. MORGANgroups decomposition where all the vertex groups are conjugate in G to subgroupsof either G1 or G2 and all the edge groups are trivial.

If this decomposition istrivial, then G′ is conjugate to a subgroup of either G1 or G2. If it is nontrivial,then either G′ is a nontrivial free product or G′ has a nontrivial free factor.□The generalization which is a natural consequence of the theory of groups actingon trees is:Theorem 5.

Let G be the fundamental group of a graph of groups, with the vertexgroups being Gv and the edge groups being Ge. Suppose that G′ ⊂G is a subgroupwith the property that its intersection with every conjugate in G of each Ge is trivial.Then G′ is the fundamental group of a graph of groups all of whose edge groupsare trivial.

In particular, G′ is isomorphic to a free product of a free group andintersections of G with various conjugates of the Gv.Proof. The restriction of the universal action of G to G′ produces an action ofG′ on a tree with trivial edge stabilizers and with vertex stabilizers exactly theintersections of G′ with the conjugates of the Gv.□Corollary 6.

With G and G′ as in Theorem 5, if G′ is not a nontrivial free product,then it is either isomorphic to Z or is conjugate to a subgroup of Gv for some vertexv.Corollary 7. With G as in Theorem 5, if G′ ⊂G is a subgroup with the propertythat the intersection of G′ with every conjugate of Gv in G is trivial, then G′ is afree group.Proof.

The point stabilizers for the universal action of G are subgroups of conju-gates of the Gv. Thus, under the hypothesis, the restriction of the universal actionto G′ is free.□These examples and applications give ample justification for the statement thatthe theory of groups acting on trees is a natural extension of classical combinatorialgroup theory.The tree associated to SL2 over a local field.

One of the main reasons thatSerre was led to investigate the theory of groups acting on trees was to constructa space to play the role for SL2(Qp) that the upper half-plane plays for SL2(R).That space is a tree. Here is the outline of Serre’s construction.Let K be a (commutative) field with a discrete valuation v: K∗→Z.

Recallthat this means that v is a homomorphism from the multiplicative group K∗of thefield onto the integers with the property thatv(x + y) ≥min(v(x), v(y)) .By convention we set v(0) = +∞. The valuation ring O(v) is the ring of {x ∈K|v(x) ≥0}.

This ring is a local ring with maximal ideal generated by any π ∈O(v)with the property that v(π) = 1. The quotient O(v)/πO(v) is called the residuefield kv of the valuation.Let W be the vector space K2 over K. The group SL2(K) of 2 × 2-matriceswith entries in K and determinant 1 is naturally the group of volume-preservingK-linear automorphisms of W.An O(v)-lattice in W is a finitely generated O(v)-submodule L ⊂W whichgenerates W as a vector space over K. Such a module is a free O(v)-module of rank

Λ-TREES AND THEIR APPLICATIONS92. We say that lattices L1 and L2 are equivalent (homothetic) if there is α ∈K∗such that L1 = α · L2.

We define the set of vertices V of a graph to be the setof homothety classes of O(v)-lattices of W. We join two vertices corresponding tohomothety classes having lattice representatives L1 and L2 related in the followingway: There is an O(v)-basis {e, f} for L1 so that {πe, f} is an O(v)-basis for L2. Itis an easy exercise to show that this defines a simplicial tree on which SL2(K) acts.The stabilizer of any vertex is a conjugate in GL2(K) of SL2(K).

The quotientgraph is the interval.A fundamental domain for the action on the tree is theinterval connecting the class of the lattice with standard basis {e, f} to the class ofthe lattice with basis {e, πf}. The stabilizer of the edge joining these two latticesis the subgroup∆=abcd∈SL2(O(v))|c ∈πO(v).Thus, we haveSL2(K) ∼= SL2(O(v)) ∗∆SL2(O(v))′whereSL2(O(v))′ =100π−1SL2(O(v))100π.If A ∈SL2(K) and if [L] ∈T , then the distance that A moves [L] is given asfollows.

We take an O(v)-basis for L and use it to express A as a 2 × 2 matrix.The absolute value of the minimum of the valuation of the 4 matrix entries is thedistance that [L] is moved. In particular, if trace (A) has negative valuation thenA fixes no point of T .Applications.

In the special case of SL2(Qp) we have a decompositionSL2(Qp) = SL2(Zp) ∗∆SL2(Zp)′ .The maximal compact subgroups of SL2(Qp) are the conjugates of SL2(Zp). Wesay that a subgroup of SL2(Qp) is discrete if its intersection with any maximalcompact subgroup is finite.Corollary 8 (Ihara’s Theorem).

Let G ⊂SL2(Qp) be a torsion-free, discrete sub-group. Then G is a free group.Proof.

If G is both torsion-free and discrete, then its intersection with any conjugateof SL2(Zp) is trivial. Applying Corollary 7 yields the result.□If C is a smooth curve, then let C(C) denote the field of rational functions on C.The valuations of C(C) are in natural one-to-one correspondence with the pointsof the completion bC of C. For each p ∈bC the associated valuation vp on C(C)is given by vp(f) is the order of the zero of f at p. (If f has a pole at p, thenby convention the order of the zero of f at p is minus the order of the pole of fat p.) Let H be a fixed finitely presented group.

The representations of H intoSL2(C) form an affine complex algebraic variety, called the representation variety,whose coordinate functions are the matrix entries of generators of H. Suppose thatwe have an algebraic curve C in this variety which is not constant on the level ofcharacters. That is to say there is h ∈H such that the trace of ρ(h) varies as ρ

10J. W. MORGANvaries in C. Let K be the function field of this curve.

Each ideal point p of C isidentified with a discrete valuation vp of K supported at infinity in the sense thatthere is a regular (polynomial) function f on C with vp(f) < 0. Because of thecondition that C be a nontrivial curve of characters, for some ideal point p, here ish ∈H such that the regular function ρ 7→trh(ρ) = trace(ρ(h)) has negative valueunder vp.Associated to such a valuation vp we have an action of SL2(K) on a simplicialtree T .

We also have the tautological representation of H into SL2(K) (in factinto SL2 of the coordinate ring of regular functions on C). (In order to define thisrepresentation, notice first that C is a family of representations of H into SL2(C).The tautological representation assigns to h ∈H the matrixf11f12f21f22,where fij(c) is the ijth entry of c(h) ∈SL2(C).) Thus, there is an induced actionof H on a simplicial tree.

If vp(trh) < 0 it follows that the element h fixes no pointof T . Thus, under our hypotheses on C and vp, it follows that the action of H onthe tree is nontrivial in the sense that H does not fix a point of the tree.According to Corollary 2 we have provedCorollary 9 ([4]).

Let H be a finitely presented group. Suppose that the charactervariety of representations of H into SL2(C) is positive dimensional.

Then there isan action of H on a tree without fixed point. In particular, H has a nontrivial de-composition as a free product with amalgamation or H has an HNN-decomposition.2.

Λ-treesIn this section we define Λ-trees as the natural generalization of simplicial trees.We prove the analogue of the result connecting trees and SL2. Namely, if K is alocal field with valuation v: K∗→Λ and if G(K) is a rank-1 group, then there isa Λ-tree on which G(K) acts.

We then take up the basics of the way groups acton Λ-trees. We classify single isometries of Λ-trees into three types and use this todefine the hyperbolic length function of an action.

This leads to a definition of thespace of all nontrivial, minimal actions of a given group on Λ-trees. We finish thesection with a brief discussion of base change in Λ.Let us begin by reformulating the notion of a simplicial tree in a way that willeasily generalize.

The vertices V of a simplicial tree T are a set with an integer-valued distance function. Namely, the distance from v0 to v1 is the path distanceor equivalently the minimal number of edges in a simplicial path in T from v0 to v1.The entire tree T can be reconstructed from the set V and the distance function.The reason is that two vertices of T are joined by an edge if and only if the distancebetween them is 1.

A question arises as to which integer-valued distance functionsarise in this manner from simplicial trees. The answer is not too hard to discover.It is based on the notion of a segment.

A Z-segment in an integer valued metricspace is a subset which is isometric to a subset of the form {t ∈Z|0 ≤t ≤n}. Theinteger n is called the length of the Z-segment.

The points corresponding to 0 andn are called the endpoints of the Z-segment.Theorem 10. Let (V, d) be an integer-valued metric space.

Then there is a sim-plicial tree T such that the path distance function of T is isometric to (V, d) if and

Λ-TREES AND THEIR APPLICATIONS11only if the following hold(a) For each v, w ∈V there is a Z-segment in V with endpoints v and w. Thissimply means that there is a sequence v = v0, v1, . .

. , vn = w such that forall i we have d(vi, vi+1) = 1.

(b) The intersection of two Z-segments with an endpoint in common is a Z-segment. (c) The union of two Z-segments in V whose intersection is a single point whichis an endpoint of each is itself a Z-segment.Given a set with such an integer-valued distance function, one constructs a graphby connecting all pairs of points at distance one from each other.

Condition (a)implies that the result is a connected graph. Condition (b) implies that it has noloops.

Condition (c) implies that the metric on V agrees with the path metric onthis tree.This definition can be generalized by replacing Z by any (totally) ordered abeliangroup Λ. Before we do this, let us make a couple of introductory remarks aboutordered abelian groups.

An ordered abelian group is an abelian group Λ which ispartitioned in three subsets P, N, {0} such that for each x ̸= 0 we have exactly oneof x, −x is contained in P and with P closed under addition. P is said to be theset of positive elements.

We say that x > y if x −y ∈P. A convex subgroup of Λ isa subgroup Λ0 with the property that if y ∈Λ0 ∩P and if 0 < x < y then x ∈Λ0.The rank of an ordered abelian group Λ is one less than the length of the maximalchain of convex subgroups, each one proper in the next.

An ordered abelian grouphas rank 1 if and only if it is isomorphic to a subgroup of R.A Λ-metric space is a pair (X, d) where X is a set and d is a Λ-distance functiond: X × X →Λ satisfying the usual metric axioms. In Λ there are segments [a, b]given by {λ ∈Λ|a ≤λ ≤b}.

More generally, in a Λ-metric space a segment is asubset isometric to some [a, b] ⊂Λ. It is said to be nondegenerate if a < b. Asbefore, each nondegenerate segment has two endpoints.Definition ([24], [13]).

A Λ-tree is a Λ-metric space (T, d) such that:(a) For each v, w ∈T there is a Λ-segment in T with endpoints v and w.(b) The intersection of two Λ-segments in T with an endpoint in common is aΛ-segment. (c) The union of two Λ-segments of T whose intersection is a single point whichis an endpoint of each is itself a Λ-segment.Example.

Let v: K∗→Λ be a possibly nondiscrete valuation.Of course, bydefinition the value group Λ is an ordered abelian group. Associated to SL2(K)there is a Λ-tree on which it acts.

The points of this Λ-tree are again homothetyclasses of O(v)-lattices in K2. The Λ-distance between two homothety classes oflattices is defined as follows.Given the classes there are representative latticesL0 ⊂L1 with quotient L0/L1 and O(v)-module of the form O(v)/αO(v) for someα ∈O(v).

The distance between the classes is v(α). In particular, the stabilizers ofvarious points will be conjugates of SL2(O(v)).

Thus, every element γ ∈SL2(K)which fixes a point in this Λ-tree has v(tr(γ)) ≥0.If ρ: H →SL2(K) is arepresentation, then there is an induced action of H on the Λ-tree. This action willbe without fixed point for the whole group if there is h ∈H such that v(tr ρ(h)) < 0.

12J. W. MorganThe tree associated to a semisimple rank-1 algebraic group over a localfield.

The example in the previous section showed how SL2 over a local field withvalue group Λ gives rise to a Λ-tree. In fact this construction generalizes to anysemisimple real rank-1 group.

We will content ourselves with considering the caseof SO(n, 1). Let K be a field with a valuation v: K∗→Λ.

We suppose that theresidue field kv is formally real in the sense that −1 is not a sum of squares inkv. (In particular, kv is of characteristic zero.) Let q: Kn+1 →K be the standardquadratic form of type (n, 1); i.e.,q(x0, .

. .

, xn) = x0x1 + x22 + · · · + x2n .We denote by ⟨·, ·⟩the induced bilinear form.Let SOK(n, 1) ⊂SLn+1(K) be the automorphism group of q. By a unimodularO(v)-lattice in Kn+1 we mean a finitely generated O(v)-module which generatesKn+1 over K and for which there is a standard O(v)-basis, i.e., an O(v)-basis{e0, .

. .

, en} with q(P yiei) = y0y1 + y22 + · · · + y2n. Let T be the set of unimodularO(v)-lattices.

To define a Λ-metric on T we need the following lemma which isproved directly.Lemma 11. If L0 and L1 are unimodular O(v)-lattices, then there is a standardO(v)-basis for L0, {e0, e1, .

. .

, en} and α ∈O(v) such that{αe0, α−1e1, e2, . .

. , en}is a standard basis for L1.We then define the distance between L0 and L1 to be v(α).

It is not too hardto show that this space of unimodular O(v)-lattices with this Λ-distance functionforms a Λ-tree (see [11]). Notice that the Λ-segment between L0 and L1 is defined bytaking the unimodular O(v)-lattices with bases {βe0, β−1e1, e2, .

. .

, en} as β rangesover elements of O(v) with v(β) ≤v(α). (This then generalizes the construction inExample 1 to SO(n, 1) and we repeat to the other rank-1 groups.

)The tree associated to SO(n, 1) over a local field K will be a simplicial treeexactly when the valuation on K is discrete (i.e., when the value group is Z).Basics of group actions on Λ-trees (cf. [13, 1, 3]).

We describe some of thebasic results about the way groups act on Λ-trees. The statements and proofs areall elementary and directly “tree-theoretic.” Let us begin by classifying a singleautomorphism α of a Λ-tree T .

There are three cases:(1) α has a fixed point in T . Then the fixed point set Fα of α is a subtree,and any point not in the fixed point set is moved by α a distance equal totwice the distance to Fα.

The midpoint of the segment joining x to α(x) iscontained in Fα. (See Figure 4.

)(2) There is a segment of length λ ∈Λ but λ /∈2Λ which is flipped by α. (SeeFigure 5.

)(3) There is an axis for α; that is to say there is an isometry from a convexsubgroup of Λ into T whose image is invariant under α and on which α actsby translation by a positive amount τ(α).

Λ-TREES AND THEIR APPLICATIONS13Figure 4Figure 5Figure 6The set of all points in T moved by α this distance τ(α) themselves form themaximal axis Aα for α. Any other point x ∈T is moved by α a distance 2d(x, Aα)+τ(α) where d(x, Aα) is the distance from x to Aα.

(See Figure 6. )The hyperbolic length l(α) of α is said to be 0 in the first two cases and τ(α) inthe last case.

The automorphism α is said to be hyperbolic if τ(α) > 0, elliptic if αhas a fixed point and to be an inversion in the remaining case. The characteristicset Aα of α is Fα in Case 1, Aα in Case 3, and empty in Case 2.

Thus, Cα is theset of points of T which are moved by α a distance equal to l(α). Clearly, there areno inversions if 2Λ = Λ.Notice that if x /∈Cα, then the segment S joining x to α(x) has the propertythat S ∩α(S) is a segment of positive length.

We denote this by saying that thedirection from α(x) toward x and the direction from α(x) toward α2(x) agree. Hereis a lemma which indicates the ‘tree’ nature of Λ-trees.Lemma 12.

Suppose that g, h are isometries of a Λ-tree T with the property thatg, h, and gh each have a fixed point in T . Then there is a common fixed point forg and h.Notice that this result is false for the plane: two rotations of opposite angleabout distinct points in the plane fail to satisfy this lemma.Proof.

Suppose that g and h have fixed points but that Fg ∩Fh = ∅. Then thereis a bridge between Fg and Fh, i.e., a segment which meets Fg in one end and Fhin the other.

Let x be the initial point of this bridge. Then the direction fromgh(x) toward x and the direction from gh(x) toward (gh)2(x) are distinct.

Thus,x ∈Cgh. Since x is not fixed by gh, it follows that gh is hyperbolic.

(See Figure 7on page 100. )□Corollary 13.

If G is a finitely generated group of automorphisms of a Λ-tree suchthat each element in G is elliptic, then there is a point of T fixed by the entire group

14J. W. MorganFigure 7G.

In particular, if 2Λ = Λ and if the hyperbolic length function of the action istrivial, then there is a point of the tree fixed by the entire group.Let us now describe some of the basic terminology in the theory of Λ-trees. LetT be a Λ-tree.

A direction from a point x ∈t is the germ of a nondegenerateΛ-segment with one endpoint being x. The point x is said to be a branch point ifthere are at least three distinct directions from x.

It is said to be a dead end ifthere is only one direction. Otherwise, x is said to be a regular point.If T has a minimal action of a countable group G, then T has no dead endsand only countable many branch points.

Each branch point has at most countablymany directions. It may well be the case when Λ = R that the branch points aredense in T .If G×T →T is an action of a group on a Λ-tree, then the function associating toeach g ∈G, its hyperbolic length, is a class function in the sense that it is constanton each conjugacy class.

We denote by C the set of conjugacy classes in G. Wedefine the hyperbolic length function of an action of G on a Λ-tree to bel: C →Λ≥0which assigns to each c ∈C the hyperbolic length of any element of G in the classc.The space of actions (cf. [3]).

Several natural questions arise. To what extentdoes the hyperbolic length function determine the action?

Which functions arehyperbolic length functions of actions? The second question has been completelyanswered (at least for subgroups of R).

There are some obvious necessary conditions(each condition being an equation or weak inequality between the hyperbolic lengthof finitely many group elements), first laid out in [3]. In [19] it was proved thatthese conditions characterize the set of hyperbolic length functions.Let us consider the first question.

The idea is that the hyperbolic length functionof an action should be like the character of a representation. There are a coupleof hurdles to surmount before this analogy can be made precise.

First of all, theinversions cause problems much like they do in the simplicial case. Thus, as in thesimplicial case, one restricts to actions without inversions (which is no restrictionat all in the case Λ = R).

One operation which changes the action but not itshyperbolic length function is to take a subtree. For most actions there is a unique

Λ-Trees and Their Applications15minimal invariant subtree. If the group is finitely generated, the actions which donot necessarily have a unique invariant subtree are those that fix points on the tree.These we call trivial actions.

It is natural to restrict to nontrivial actions and towork with the minimal invariant subtree. In this context the question then becomesto what extent the minimal invariant subtree of a nontrivial action is determined bythe hyperbolic length function of the action.

There is a special case of ‘reducible-type’ actions much like the case of reducible representations where the hyperboliclength function does not contain all the information about the minimal invariantsubtree, but this case is the exception. For all other actions one can reconstructthe minimal action from the hyperbolic length function.

These considerations leadto a space of nontrivial, minimal actions of a given finitely presented group on Λ-trees. It is the space of hyperbolic length functions.

When Λ has a topology, e.g.,if Λ ⊂R then this space of actions inherits a topology from the natural topologyon the set of Λ-valued functions on the group. For example, when Λ = R the spaceof nontrivial, minimal actions is closed subspace of (R≥0)C −{0}.

It is natural todivide this space by the action of R+ by homotheties forming a projective spaceP((R≥0)C) = ((Rge0)C −{0}/R+) .This projective space is compact, and the space of projective classes of actions (orprojectivized hyperbolic length functions) is a closed subset of this projective spacePA(G) ⊂P((R≥0)C).Base change (cf. [1]).

Suppose that Λ ⊂Λ′ is an inclusion of ordered abeliangroups. Suppose that T is a Λ-tree.

Then there is an extended tree T ⊗Λ Λ′. Inbrief one replaces each Λ-segment in T with a Λ′-segment.

One example of thisis T ⊗Z R. This operation takes a Z tree (which is really the set of vertices of asimplicial tree) and replaces it with the R-tree, which is the geometric realizationof the simplicial tree. The operation T ⊗Z Z[1/2] is the operation of barycentricsubdivision.

Base change does not change the hyperbolic length function.If G acts on a Λ-tree T , then it acts without inversions on the Λ[1/2]-tree T ⊗ΛΛ[1/2].The operation of base change is a special case of a more general constructionthat embeds Λ-metric spaces satisfying a certain 4-point property isometrically intoΛ-trees; see [1].Finally, if Λ′ is a quotient of Λ by a convex subgroup Λ0, then there is ananalogous quotient operation that applies to any Λ tree to produce a Λ′-tree asquotient. The fibers of the quotient map are Λ0-trees.3.

Compactifying the space of charactersLet G be a finitely presented group. R(G) = Hom(G,SO(n,1)) is naturally thereal points of an affine algebraic variety defined over Z.

In fact, if {g1, . .

. , gk} aregenerators for G then we haveHom(G, SO(n, 1)) ⊂SO(n, 1)k ⊂M(n × n)k = Rkn2is given by the polynomial equations which say that (1) all the gi are mapped toelements of SO(n, 1) and (2) that all the relations among the {gi} which hold in

16J. W. MORGANG hold for their images in SO(n, 1).We denote by SOC(n, 1) and RC(G) thecomplex versions of these objects.

Each g ∈G determines n2 polynomial functionson R(G). These functions assign to each representation the matrix entries of therepresentation on the given element g. The coordinate ring for R(G) is generatedby these functions.

These is an action of SOC(n, 1) and RC(G) by conjugation.The quotient affine algebraic variety is called the character variety and is denotedχC(G) Though it is a complex variety, it is defined over R. Its set of real points,χ(G), is the equivalence classes of complex representations with real characters.The polynomial functions on χ(G) are the polynomial functions on R(G) invariantby conjugation. In particular the traces of the various elements of G are polynomialson χ(G).

The character variety contains a subspace Z(G) of equivalence classes ofreal representations. The subspace Z(G) is a semialgebraic subset of χ(G) and isclosed in the classical topology.

Each fiber of the map R(G) →Z(G) either is madeup of representations whose images are contained in parabolic subgroups of SO(n, 1)or is made up of finitely many SO(n, 1)-conjugacy classes of representations intoSO(n, 1).The variety χ(G) and the subspace Z(G) are usually not compact. Our purposehere is produce natural compactifications of Z(G) and χ(G) as topological spacesand to interpret the ideal points at infinity.The idea is to map the charactervariety into a projective space whose homogeneous coordinates are indexed by C,the set of conjugacy classes in G. The map sends a representation to the pointwhose homogeneous coordinates are the logs of the absolute values of its traces onthe conjugacy classes.

The first result is that the image of this map has compactclosure in the projective space. As we go offto infinity in the character varietyat least one of these traces is going to infinity.

Thus, the point that we convergeto in the projective space measures the relative growth rates of the logs of thetraces of the various conjugacy classes. It turns out that any such limit point inthe projective space can be described by a valuation supported at infinity on thecharacter variety (or on some subvariety of it).

These then can be reinterpretedin terms of actions of G on R-trees. This then is the statement for the case ofSO(n, 1): A sequence of representations of G into SO(n, 1) which has unboundedcharacters has a subsequence that converges to an action of G on an R-tree.

Thismaterial is explained in more detail in [13] and [11].The projective space. We begin by describing the projective space in which weshall work.

Let C denote the set of conjugacy classes in G. We denote by P(C) theprojective space((R≥0)C −{0})/R+where R+ acts by homotheties. Thus, a point in P(C) has homogeneous coordinates[xγ]γ∈C with the convention that each xγ is a nonnegative number.The main result.

Before we broach all the technical details, let us give a conse-quence which should serve to motivate the discussion.Theorem 14. Let ρk : G →SO(n, 1) be a sequence of representations with theproperty that for some g ∈G we have {tr(ρk(g)}k is unbounded.

Then after replac-ing the {ρk} with a subsequence we can find a nontrivial actionϕ: G × T →T

Λ-TREES AND THEIR APPLICATIONS17of G on an R-tree such that the positive part of the logs of the absolute value of thetraces of the ρk(g) converge projectively to the hyperbolic length function of ϕ; i.e.,so that ifpk = [max(0, log | tr(ρk(γ))|)]γ∈C ∈P(C)thenlimk7→∞pk = [l(ϕ(γ))]γ∈C .If all the representations ρk of G into SO(n, 1) are discrete and faithful, then thelimiting action of G on an R-tree has the property that for any nondegeneratesegment J ⊂T the stabilizer of J under ϕ is a virtually abelian group.The component of the identity SO+(n, 1) of SO(n, 1) is an isometry groupof hyperbolic n-space.We define the hyperbolic length of α ∈SO(n, 1) to bethe minimum distance a point in hyperbolic space is moved by α. The quantitymax(0, log(| tr(α)|)) differs from the hyperbolic length of α by an amount boundedindependent of α.

Expressed vaguely, the above result says that as actions of G onhyperbolic n-space degenerate the hyperbolic lengths of these actions, after rescal-ing, converge to the hyperbolic length of an action of G on an R-tree.The rest of this section is devoted to indicating how one establishes this result.Mapping valuations into P(C). Let us describe how valuations on the functionfield of R(G) determine points of P(C).

Suppose that Λ ⊂R. Then any collectionof elements {xγ}γ∈C of Λ≥0, not all of which are zero, determine an element ofP(C).

This can be generalized to any ordered abelian group of finite rank. Supposethat Λ is an ordered abelian group and that x, y ∈Λ are nonnegative elements, notboth of which are zero.

Then there is a well-defined ratio x/y ∈R≥0 ∪∞. Moregenerally, if {xγ}γ∈C is a collection of elements in Λ with the property that xγ ≥0for all γ ∈C and that xγ > 0 for some γ ∈C and if Λ is of finite rank, then we havea well-defined point[xγ]γ∈C ∈P(C) .Now suppose that K is the function field of R(G).

This field contains the tracefunctions trγ for all γ ∈C. A sequence in χ(G) goes offto infinity if and only if atleast one of the traces trγ is unbounded on the sequence.Lemma 15.

Let v: K∗→Λ is a valuation, trivial on the constant functions, whichis supported at infinity in the sense that for some γ ∈C we have v(trγ) < 0. ThenΛ is of finite rank.

Thus we can define a point µ(v) in P(C) byµ(v) = [max(0, −v(trγ))]γ∈C .Similarly, if X ⊂R(G) is a subvariety defined over Q and if v is a valuation on itsfunction field KX, trivial on the constant functions, supported at infinity then wecan define µ(v) to be the same formula.The reason that we take max(0, −v(trγ)) is that this number is the logarithmicgrowth of trγ as measured by v.For example, if the field is the function fieldof a curve and v is a valuation supported at an ideal point p of the curve thenmax(0, −v(f)) is exactly the order of pole of f at p.

18J. W. MORGANHow actions of G on Λ-trees determine points in P(C).

Another source ofpoints in P(C) is nontrivial actions of G on Λ-trees, once again for Λ an orderedabelian group of finite rank. Let ϕ: G × T →T be an action of G on a Λ-tree.

LetT ′ be the Λ[1/2]-tree T ⊗Λ Λ[1/2]. Then there is an extended action of G on T ′which is without inversions.

Since G is finitely generated, this action is nontrivialif and only if its hyperbolic length function l: C →Λ≥0 is nonzero. If the action isnontrivial, then we have the pointl(ϕ) = [l(ϕ(γ))]γ∈C ∈P(C) .Actually, it is possible to realize the same point in P(C) by an action of G on anR-tree.

The reason is that we can find a convex subgroup Λ0 ⊂Λ which contains allthe hyperbolic lengths and is minimal with respect to this property. Let Λ1 ⊂Λ0be the maximal proper convex subgroup of Λ0.

Then the T admits a G-invariantΛ0-subtree. This tree has a quotient Λ0/Λ1-tree on which G acts.

Since Λ0/Λ1 isof rank 1, it embeds as a subgroup of R. Thus, by base change we can extend theG action on this quotient tree to a G action on R. This action determines the samepoint in P(C).As we saw in the last section, the projective space PA(G) of nontrivial minimalactions of G on R-trees sits by definition P(C). What we have done here is to showthat any action of G on a Λ-tree, for Λ an ordered abelian group of finite rank, alsodetermines a point of PA(G) ⊂P(C).There is a relationship between these two constructions of points in P(C), onefrom valuations on the function field of χ(G) and the other from actions on trees.Theorem 16.

Suppose that X ⊂R(G) is a subvariety defined over Q and whoseprojection to χ(G) is unbounded. Suppose that KX is its function field and thatv: K∗X →Λ is a valuation, trivial on the constant functions, which is supportedat infinity and which has formally real residue field.

Then associated to v is anaction of SOK(n, 1) on a Λ-tree. We have the tautological representation of G intoSOK(n, 1) and hence there is an induced action ϕ of G on a Λ-tree.

The imagein P(C) of the valuation and of this action of G on the tree are the same; i.e.,µ(v) = l(ϕ).Embedding the character variety in P(C). As we indicated in the beginningof this section, the purpose for introducing the maps from valuations and actionson trees to P(C) is to give representatives for the ideal points of a compactificationof the character variety, χ(G).

Toward this end we define a map θ: χ(G) →P(G)byθ([ρ]) = [max(0, log(| trγ(ρ)|))]γ∈C .The map θ measures the relative sizes of the logs of the absolute values of thetraces of the images of the various conjugacy classes under the representation. Itis an elementary theorem (see [13]) that the image θ(χ(G)) has compact closure.Thus, there is an induced compactification χ(G).

The set of ideal points of thiscompactification, denoted by B(χ(G)), is the compact subset of points in P(C)which are limits of sequences {θ([ρi])}i where {[ρi]}i is an unbounded sequence inχ(G). We denote by B(Z(G)) the intersection of the closure of Z(G) with B(χ(G)).Here are the two main results that were established in [13] and [11].

Λ-TREES AND THEIR APPLICATIONS19Theorem 17. For each point b ∈B(χ(G)) there exist a subvariety X of R(G)defined over Q and a valuation v on KX, trivial on the constant functions, supportedat infinity such that µ(v) = b.Theorem 18.

For each point b ∈B(Z(G)) the valuation v in Theorem 17 withµ(v) = b can be chosen to have formally real residue field. Thus, there is a nontrivialaction ϕ: G × T →T on an R-tree such that b = l(ϕ).

Said another way B(Z(G))is a subset of PA(G) ⊂P(C). Lastly, if the point b ∈B(Z(G)) is the limit ofdiscrete and faithful representations, then the action ϕ has the property that thestabilizer of any nondegenerate segment in the tree is a virtually abelian subgroupof G.Theorem 14 is now an immediate consequence of this result.There is an obvious corollary.Corollary 19.

Let G be a finitely presented group nonvirtually abelian group. Ifthere is no nontrivial action of G on an R-tree in which the stabilizer of eachnondegenerate segment is virtually abelian, then the space of conjugacy classes ofdiscrete and faithful representations of G into SO(n, 1) is compact.What is not clear in this result, however, is the meaning of the condition about Gadmitting no nontrivial actions on R-trees with all nondegenerate segments havingvirtually abelian stabilizers.

This is a question to which we shall return in §5.One thing is clear from the algebraic discussion, however. The discrete valuationsof a field are dense in the space of all valuations on the field.

Thus, it turns outthat a dense subset of B(χ(G)) is represented by discrete valuations. This leads tothe following result.Proposition 20.

B(Z(G)) contains a countable dense subset represented by non-trivial actions of G on simplicial trees.From this result and Corollary 2 we haveCorollary 21. Let G be a finitely generated group.

If Z(G) is not compact ; i.e.,if there is a sequence of representations {ρk}k of G into SO(n, 1) such that forsome γ ∈G the sequences of traces {tr(ρk(γ))}k is unbounded, then G has ei-ther a nontrivial free product with amalgamation decomposition or G has an HNN-decomposition.Geometric approach. What we have given here is an algebraic approach to estab-lish the degeneration of actions of G on hyperbolic space to actions of G on R-trees.There is, however, a purely geometric approach to the same theory.

In a geometricsense one can take limits of hyperbolic space with a sequence of rescaled metricsso that the curvature goes to −∞. Any such limit is an R-tree.

Given a finitelygenerated group, there is a finite set of elements {g1, . .

. , gt} in G such that forany g ∈G there is a polynomial pg in t variables such that for any representationρ: G →SO(n, 1) the trace of ρ(g) is bounded by the value pg(ρ(g1), .

. .

, ρ(gt)).Thus, given a sequence of representations ρk then for each k we replace hyper-bolic space by a constant λk so that the maximum of the hyperbolic lengths ofρk(g1), . .

. , ρk(gt) is 1.

Any limit of these rescaled hyperbolic spaces will be anR-tree on which there is an action of G. For more details see [12, §§8–10].

20J. W. MORGANFigure 84.

Trees and codimensional-1 measured laminationsIn this section we shall explain the relation between trees and codimension-1transversely measured laminations. This relationship generalizes Example 4 of §1where it was shown that if N ⊂M is a compact codimension-1 submanifold, thenthere is a dual action of π1(M) on a simplicial tree.Definition.

Let M be a manifold. A (codimension-1 transversely) measured lam-ination (L, µ) in M consists of(a) a closed subset |L| ⊂M called the support of the lamination L,(b) a covering of |L| by open subsets V of M, called flow boxes, which havetopological product structures V = U × (a, b) (where (a, b) denotes an openinterval) so that |L| ∩V is of the form U × X where X ⊂(a, b) is a closedsubset, and(c) for each open set as in (b) a Borel measure on the interval (a, b) with supportequal to X ⊂(a, b).The open sets and measures are required to satisfy a compatibility condition.We define the local leaves of the lamination in V = U × (a, b) to be the slicesU × {x} for x ∈X.

The first compatibility condition is that the germs of localleaves in overlapping flow boxes agree. The second compatibility condition involvestransverse paths.

A path in a flow box is said to transverse to the lamination if it istransverse to each local leaf. Then the measures in the flow box can be integratedover transverse paths in that flow box to give a total measure.

If a transverse pathlies in the intersection of two flow boxes then the total measures that are assignedto it in each flow box are required to agree.Two sets of flow boxes covering |L| define the same structure if their unionforms a compatible system of flow boxes. Equivalently, we can view a measuredlamination as a maximal family of compatible flow boxes.We define an equivalence relation on |L|.

This equivalence relation is generatedby saying that two points are equivalent if they both lie on the same local leafin some flow box. The equivalence classes are called the leaves of the measuredlamination.

Each leaf is a connected codimension-1 submanifold immersed in a1-to-1 fashion in M. It meets each flow box in a countable union of local leaves forthat flow box. (See Figure 8.

)If the ambient manifold is compact, then the cross section of the support of ameasured lamination is the union of a cantor set where the measure is diffuse,an isolated set where the measure has δ-masses and nondegenerate intervals. If thelamination has no compact leaves, then every cross section meets the support in a

Λ-Trees and Their Applications21Figure 9cantor set.The complement of the support of a measured lamination is an open subset ofM. It consists then of at most countably many components.

It turns out that whenM is compact, there can be countably many ‘thin’ or product components boundedby two parallel leaves. The other complementary components are finite in numberand are called the ‘big’ complementary regions.

(See Figure 9. )Example 1.

A compact, codimension-1 submanifold N ⊂M is a codimension-1measured lamination; we assign the counting measure, that is to say a δ-mass oftotal mass 1 transverse to each component of N.Example 2. Thurston [23] has considered measured laminations on a closed hy-perbolic surface a genus g all of whose leaves are geodesics.

He has shown thatthe space of all such measured laminations with the topology induced from theweak topology on measures is a real vector space of dimension 6g −6. The typicalgeodesic measured lamination has 4g −4 complementary regions each of which isan ideal triangle.

(See Figure 9. )It is always possible to thicken up any leaves that support δ-masses for thetransverse measure to a parallel family of leaves with diffuse measure.Let usassume that we have performed this operation, so that our measured laminationshave no δ-masses.Here is one theorem which relates actions on trees with measured laminations.Theorem 22 [16].

Let M be a compact manifold, and let (L,µ) be a measuredlamination in M. Suppose the following hold :(1) The leaves of the covering eL in the universal covering fM of M are propersubmanifolds. (2) If x, y ∈fM, then there is a path joining them which is transverse to eL andwhich meets each leaf of eL at most once.Then there is a dual R-tree TL to eL and an action of π1(M) on TL.

The branchpoints of TL correspond to the big complementary regions of eL in fM, and thusthere are only finitely many branch points modulo the action of π1(M). The regularpoints of TL correspond to the leaves of L which are not boundary components of“big” complementary regions.There is a continuous, π1(M) equivariant map fM →TL such that the inverseimage of each branch point is a single complementary region and the preimage of

22J. W. Morganeach regular point is either a single leaf of the two leaves bounding a thin comple-mentary component.In this way we see that “most” measured laminations give rise to dual actions ofthe fundamental group on R-trees.

There is also a partial converse to this result.Given an action of π1(M) on an R-tree T it is possible to construct a transverse,π1(M)-equivariant map from the universal covering fM of M to T . Using this map,one can pull back a measured lamination from the tree and the metric on it.

Thus,one produces a measured lamination associated to the action on the tree. Thismeasured lamination dominates the original action.

For example, if the laminationsatisfies the hypothesis of Theorem 22, then dual to it there is another action ofπ1(M) on an R-tree. This new action maps in an equivariant manner to the originalaction.

If the original action is minimal, then this map will be onto, and hence thenew action dominates the original one in a precise sense.There are several difficulties with this construction. First, it is not known thatone can always do the construction so that the resulting lamination is dual to anaction on a tree.

Second, this construction is not unique; there are many choices ofthe transverse map. In many circumstances one hopes to find a “best” transversemap, but this can be hard to achieve.Let M be a compact manifold.

We say that an action of π1(M) on an R-treeis geometric for M if there is a measured lamination (L, µ) in M which satisfiesthe hypothesis of Theorem 22 and such that the action of π1(M) dual to (L, µ)is isomorphic to the given action. We say that an action of an abstract finitelypresented group G is geometric if there is a compact manifold M with π1(M) = Gand with the action geometric for M. One question arises: Is every minimal actionof a finitely presented group geometric?

If so it would follow that for a minimalaction of a finitely presented group G on an R-tree there are only finitely manybranch points modulo the action of G and only finitely many directions from anypoint x of the tree modulo Gx. It is not known whether these statements are truein general.In spite of these difficulties, measured laminations are an important tool forstudying actions of finitely presented groups on trees.

One extremely importantresult for measured laminations which gives a clue as to the dynamics of actions onR-trees is the following decomposition result.Theorem 23 (cf. [13]).

Let (L, µ) be a measured lamination in a compact manifoldM. Then there is a decomposition of L into finitely many disjoint sublaminationseach of which has support which is both open and closed in |L|.

Each of the sub-laminations is of one of the following three types :(i) a parallel family of compact leaves,(ii) a twisted family of compact leaves with central member having nontrivialnormal bundle in M and all the other leaves being two-sheeted sections ofthis normal bundle, and(iii) a lamination in which every leaf is dense.We call a lamination of the last type an exceptional minimal lamination.One wonders if there is an analogous decomposition for actions of finitely pre-sented groups on R-trees.

Λ-Trees and Their Applications235. Applications and questionsIn the analogy we have been developing between the theories of groups actingon simplicial trees and of groups acting on more general trees, there is one miss-ing ingredient.There is no analogue for the combinatorial group theory in thesimplicial case.There are some results that should be viewed as partial steps inthis direction.

In this section, we shall discuss some of these results and give ap-plications of them to the spaces of hyperbolic structures on groups. Two thingsemerge, clearly, from this discussion.

First of all, information about the combinato-rial group theoretic consequences of the existence of an action of G on a Λ-tree hassignificant consequences for the space of hyperbolic structures on G. Second, mostof the combinatorial group theoretic consequences to date have followed from theuse of measured laminations and related ergodic considerations: first return maps,interval exchanges, etc.Applications to hyperbolic geometry. Let us begin by formalizing the notionof a hyperbolic structure.

Let G be a nonvirtually abelian, finitely generated group.For each n we denote by Hn(G) the space of hyperbolic structures on G.Bydefinition this means the conjugacy classes of discrete and faithful representationsof G into the isometry group of hyperbolic n-space. Thought of another way apoint in Hn(G) is a complete Riemannian n-manifold with all sectional curvaturesequal to −1 and with fundamental group identified with G.Example 1.

Let G be the fundamental group of a closed surface of genus at least 2.Then H2(G) is the classical Teichm¨uller space. It is homeomorphic to a Euclideanspace of dimension 6g −6.Example 2.

Let G be the fundamental group of a closed hyperbolic manifold ofdimension n ≥3. According to Mostow rigidity [18], Hn(G) consists of a singlepoint.Example 3.

Let G be the fundamental group of a closed hyperbolic manifold ofdimension n. Then Hn+1(G) can be of positive dimension (see [9]).The material in §3 leads to the following result:Theorem 24 ([13, 11]). There is a natural compactification of Hn(G).

Each idealpoint of this compactification is represented by a nontrivial action of G on R-treeswith property that the stabilizer of every nondegenerate segment of the tree is avirtually abelian subgroup of G.In the special case of surfaces this gives a compactification of the Tiechm¨ullerspace of a surface of genus ≥2 by a space of actions of the fundamental groupon R-trees. It turns out that all these actions are geometric for the surface; i.e.,the points at infinity in this compactification can be viewed as geodesic measuredlaminations.

For more details see [22] and [13].The Teichm¨uller space of a surface is acted on properly discontinuously by themapping class group (the group of outer automorphisms of the fundamental group ofthe surface). The action of the mapping class group extends to the compactificationof Teichm¨uller.

From this one can deduce various cohomological results for themapping class group that make it look similar to an algebraic group. Motivated bythis result, Culler-Vogtmann [5] have defined a space of free, properly discontinuousactions of a free group on R-trees.

This space is the analogue of the Teichm¨uller

24J. W. MORGANspace for the outer automorphism group of the free group.

The action of the outerautomorphism group of the free group on this space extends to the compactificationof this space of free, properly discontinuous actions inside the projective space ofall actions of the free group on R-trees. By [3] the limit points are actions of thefree group on R-trees with stabilizers of all nondegenerate segments being virtuallyabelian.

Using this action, Culler-Vogtmann deduce many cohomological propertiesof the outer automorphism group of a free group.Theorem 24 leads immediately to the question: Which groups act nontrivially onR-trees with the stabilizer of every nondegenerate segment in the tree being virtuallyabelian? The answer for simplicial trees follows immediately from the analysis in§1.Proposition 25.

A group G acts nontrivially on a simplicial tree with all edgestabilizers being virtually abelian if and only if G has either(A) a nontrivial free production with amalgamation with the amalgamating groupbeing virtually abelian, or(B) an HNN-decomposition with the subgroup being virtually abelian.For fundamental groups of low dimensional manifolds, the answer is also known.This is a consequence of a study of measured laminations in 3-manifolds and, inparticular, the ability to do surgery on measured laminations in 3-manifolds tomake them incompressible.Theorem 26 ([14, 15]). If M is a 3-manifold, then π1(M) acts nontrivially on anR-tree with the stabilizers of all nondegenerate segments being virtually abelian ifand only if it has such an action on a simplicial tree.As an application we haveCorollary 27.

Let G be the fundamental group of a compact 3-manifold M. Thenfor all n the space Hn(G) is compact unless G has a decomposition of type (A) or(B) in Proposition 25. In particular, if the interior of M has a complete hyperbolicstructure and has incompressible boundary, then Hn(G) is noncompact if and onlyif M has an essential annulus which is not parallel into ∂M.One suspects that the first statement is true without the assumption that G isa 3-manifold group.Conjecture.

Let G be a finitely presented group. Then for all n the space Hn(G)is compact unless G has a decomposition of type (A) or (B) in Proposition 25.One also suspects that any action of G on an R-tree with segment stabilizersbeing virtually abelian can be approximated by an action of G on a simplicial treewith the same property.

This is known for surface groups by [22], but not in general.The fundamental group of a closed hyperbolic k-manifold, for k ≥3, has nodecomposition of type (A) or (B). In line with the above suspicions we have aresult proved by the geometric rather than algebraic means.Theorem 28 ([12]).

Suppose that G is the fundamental group of a closed hyperbolicmanifold of dimension k ≥3. Then for all n, the space Hn(G) is compact.Combinatorial group theory for groups acting on R-trees.

Perhaps thesimplest question about the combinatorial analogues of the simplicial case is thefollowing: which (finitely presented) groups act freely on R-trees? Another question

Λ-TREES AND THEIR APPLICATIONS25of importance, suggested by the situation for valuations is: Can one approximatea nontrivial action of a group G on an R-tree by a nontrivial action of G on asimplicial tree? If the original action has stabilizers of all nondegenerate segmentsbeing virtually abelian, it there a simplicial approximation with this property?

(Forexample, does the boundary of Culler-Vogtmann space have a dense subset consistingof simplicial actions? )We finish by indicating some of the partial results concerning these two ques-tions.Because of the existence of geodesic laminations with simply connectedcomplementary regions, surface groups act freely on R-trees (see [16]).

Of course,any subgroup of R also acts freely on an R-tree. It follows easily that any freeproduct of these groups acts freely on an R-tree.

The question is whether thereare any other groups which act freely. By Theorem 26, for 3-manifold groups theanswer is no.We say that a group is indecomposable if it is not a nontrivial free product.Clearly, it suffices to classify indecomposable groups which act freely on R-trees.In [10] and [17] the class of finitely presented, indecomposable groups which have anontrivial decomposition as a free product with amalgamation or HNN-decomposition,where in each case the subgroup is required to be virtually abelian, were studied.

Itwas shown that if such a group acts freely on an R-tree, then it is either a surfacegroup or a free abelian group. This result is proved by invoking the theory of mea-sured laminations and results from ergodic theory.

Another result which followsfrom using the same techniques is: A minimal free action of an indecomposable,noncyclic group G on an R-tree T is mixing in the following sense. Given any twonondegenerate segments I and J in T , there is a decomposition I = I1 ∪· · · ∪Ipand group elements gi ∈G such that for all i we have gi · Ii ⊂J.

In particular, theorbit G · J of J is the entire tree.In a different direction, in [6] Gillet-Shalen proved that if G acts freely on anR-tree which is induced by base change from a Λ-tree where Λ is a subgroup of Rwhich generates a rational vector space of rank at most 2, then G is a free productof surface groups and free abelian groups. By the same techniques they, togetherwith Skora [7], show that actions on such Λ-trees can be approximated by actionson simplicial trees (preserving the virtually abelian segment stabilizer condition ofrelevant).Recently, Rips has claimed that the only finitely generated groups which actfreely on R-trees are free products of surface groups and free abelian groups.References1.

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94 (1988), 605–622.11., Group actions on trees and the compactification of the space of classes of SO(n, 1)-representations, Topology 25 (1986), 1–33.12., Trees and degenerations of hyperbolic structures, CBMS Lecture Note Series (toappear).13. J. Morgan and P. Shalen, Valuations, trees, and degenerations of hyperbolic structures, I,Ann.

of Math. (2) 120 (1984), 401–476.14., Valuations, trees, and degenerations and hyperbolic structures, II: measured lam-inations in 3-manifolds, Ann.

of Math. (2) 127 (1988), 403–465.15., Valuations, trees, and degenerations of hyperbolic structures, III: action of 3-manifold groups on trees and Thurson’s compactness theorem, Ann.

of Math. (2) 127(1988), 467–519.16., Free actions of surface groups on R-trees, Topology (to appear).17.

J. Morgan and R. Skora, Groups acting freely on R-trees, preprint.18. G. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic spaceforms, Inst.

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295–330.20. J.-P. Serre, Trees, Springer-Verlag, New York, 1980.21.

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W. Thurston, Geometry and topology of 3-manifolds, Princeton Univ., preprint, 1980.24. J. Tits, A theorem of Lie-Kolchin for trees, Contributions to Algebra: A Collection ofPapers Dedicated to Ellis Kolchin, Academic Press, New York, 1977, pp.

377–388.Department of Mathematics, Columbia University, New York, New York 10027E-mail address: jm@math.columbia.edu

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