$\R$-trees and laminations for free groups III: Currents and dual $\R$-tree metrics
aa r X i v : . [ m a t h . G R ] J un R -trees and laminations for free groups III:Currents and dual R -tree metrics Thierry Coulbois, Arnaud Hilion and Martin LustigMay 28, 2018
A geodesic lamination L on a closed hyperbolic surface S , when provided witha transverse measure µ , gives rise to a “dual R -tree” T µ , together with anaction of G = π S on T µ by isometries. A point of T µ corresponds precisely toa leaf of the lift e L of L to the universal covering e S of S , or to a complementarycomponent of e L in e S . The G -action on T is induced by the G -action on e S as deck transformations. This construction is well known (see [Mor86]). Itis also known [Sko96] that conversely, for every small isometric action of asurface group G = π S on a minimal R -tree T there exists a “dual” measuredlamination ( L , µ ) on S , i.e. one has T = T µ up to a G -equivariant isometry.This beautiful correspondence has tempted geometers and group theoriststo investigate possible generalizations, and the first one arises if one replacesthe closed surface by a surface with boundary, and correspondingly the sur-face group G by a free group F N of finite rank N ≥
2. A first glimpse ofthe potential problems can be obtained from two simultaneous but distinctidentifications F N ∼ = −→ π S and F N ∼ = −→ π S , thus obtaining actions of π S on a tree T which are dual to a measured lamination on S , but in generalnot dual to any measured lamination on the surface S .Worse, using the index of an R -tree action by F N as introduced in [GL95],it is easily seen that for many (perhaps even “most”) small or very small R -trees T with isometric F N -action there is no identification whatsoever of F N with the fundamental group of any surface that would make T dual toa lamination. An example of such trees are the forward limit trees T α of1ertain irreducible automorphisms with irreducible powers (so called iwip automorphisms) of F N . Much like pseudo-Anosov surface homeomorphisms,such an iwip automorphism has precisely one forward and one backward limittree, T α and T α − respectively, and it induces a North-South dynamics on thespace CV N of projectivized very small F N -actions on R -trees (see [LL03]).Note that, contrary to the case of pseudo-Anosov homeomorphisms, it is afrequent occurence for an iwip automorphism α (see Corollary 5.7 below)that its stretching factor λ α is different from the stretching factor λ α − of itsinverse.In [CHL-II] for any R -tree T with isometric F N -action, a dual lamination L ( T ) has been defined, which is the generalization of the geodesic lamination L for a surface tree T µ as discussed above. The goal of the present paperis to investigate the effect of putting an invariant measure µ on the duallamination L ( T ), or, in the proper technical terms, considering a free groupcurrent µ with support contained in L ( T ). We prove here, if the F N -actionon T is very small and has dense orbits, that such a current defines indeedan induced measure on the metric completion T of T .In the special case considered above where T = T µ is dual to a measuredlamination ( L , µ ) on a surface, then the transverse measure µ defines indeeda current on L ( T µ ), and the induced measure on T µ defines a dual distance on T µ which is precisely the same as the original distance on T µ (i.e. the one thatcomes from the transverse measure µ on L ). For arbitrary very small trees T with dense F N -orbits, the measure on T induced by a current µ on L ( T )defines also a metric on T , except that this dual metric d µ may in generalbe in various ways degenerate (compare § T . Alternatively,it could be zero throughout the interior T of T .The main result of this paper is to show that these “exotic” phenom-ena are not just theoretically possible, but that they actually do occur inimportant classes of examples.Let α ∈ Aut ( F N ) be an iwip automorphism, let T α be the forward limittree of α . Then the dual lamination L ( T α ) is uniquely ergodic (see Proposi-tion 5.6): it carries a projectively unique non-trivial current µ . In this casethe dual metric d µ is simply called the dual distance d ∗ on T α or on T α . Theorem 1.1.
Let α ∈ Aut ( F N ) be an iwip automorphism with λ α = λ α − .Then the dual distance d ∗ on the forward limit tree T α is zero or infinitethroughout T α . µ , the support Supp( µ ). The latter belongs to the spaceΛ( F N ) of laminations for the free group F N , which has been defined andinvestigated in detail in [CHL-I]. This gives a rather natural map from thespace Curr( F N ) of currents to the space Λ( F N ). The space Curr( F N ) of cur-rents µ , as well as the resulting compact space P Curr( F N ) of projectivizedcurrents [ µ ], admit a natural action of the group Out( F N ) of outer automor-phisms of F N . The results derived in Proposition 3.1 and in Lemmas 3.2, 3.3and 3.4 can be summarized as follows: Theorem 1.2.
The map Supp : Curr ( F N ) → Λ( F N ) induces a map P Supp : P Curr ( F N ) → Λ( F N ) which has the following properties:1. P Supp is Out ( F N ) -invariant.2. P Supp is not injective.3. P Supp is not surjective. However, every lamination L ∈ Λ( F N ) pos-sesses a sublamination L ⊂ L which belongs to the image of the map P Supp.4. P Supp is not continuous. However, if ( µ n ) n ∈ N is a sequence of currentswhich converges to a current µ , then the sequence of algebraic lamina-tions P Supp ([ µ n ]) has at least one accumulation point in Λ( F N ) , andany such accumulation point is a sublamination of P Supp ([ µ ]) . R -trees andmeasured laminations, as known from the surface situation, does not fullyextend to the world of free groups, very small R -tree actions and currents.Unexpected degenerations seem to occure almost as a rule, and much furtherresearch is needed before one can speak of a “true understanding”.On the other hand, the spaces of currents, of R -tree actions, and of alge-braic laminations for F N are naturally related, and although this relationshipis more challenging than in the surface case, there is clearly enough interest-ing structure there to justify further research efforts. A small such furthercontribution has already been given, in [CHL05], where algebraic laminationswhere used to characterize R -trees up to F N -equivariant variations of theirmetric. Acknowledgements.
This paper originates from a workshop organized atthe CIRM in April 05, and it has greatly benefited from the discussions startedthere and continued around the weekly Marseille seminar “Teichm¨uller” (par-tially supported by the FRUMAM). F N Let A be a basis of the free group F N or finite rank N ≥
2, and let F ( A )denote the set of finite reduced words in A ± , which is as usually identifiedwith F N . A geodesic current for a free group F N can be defined in variousways. In particular, there are the following three equivalent ways to introducethem: I. Symbolic dynamist’s choice:
Consider the space Σ A of biinfinite re-duced indexed words Z = . . . z i − z i z i +1 . . . in A ± , provided with the producttopology, the shift operator σ : Σ A → Σ A , and with the involution Z Z − ,see [CHL-I]. A geodesic current is a non-trivial σ -invariant finite Borel mea-sure µ on Σ A . We also require that µ is symmetric : the measure is preservedby the involution of Σ A given by the inversion Z Z − . II. Geometric group theorist’s choice:
Consider the space ∂ F N of pairs( X, Y ) of boundary points X = Y ∈ ∂F N , endowed with the “product” topol-ogy, with the canonical diagonal action of F N , and with the flip involution4 X, Y ) ( Y, X ) as specified in [CHL-I]. A geodesic current is a non-trivial F N - and flip-invariant Radon measure µ on ∂ F N , i.e. a Borel measure thatis finite on any compact set. III. Combinatorist’s choice:
A geodesic current is given by a non-zerofunction µ : F N = F ( A ) → R ≥ with µ ( w − ) = µ ( w ) for all w ∈ F ( A ),which satisfies the left and the right Kolmogorov property : For all reducedwords w = y . . . y k ∈ F ( A ) one has µ ( w ) = X y ∈A∪A − r { y − k } µ ( wy ) = X y ∈A∪A − r { y − } µ ( yw ) . This three viewpoints correspond to the three equivalent definitions givenin [CHL-I] of a lamination for the free group F N . We assume some famil-iarity of the reader with these three settings and will freely consider that alamination is altogether symbolic (viewpoint I), algebraic (viewpoint II) and,a laminary language (viewpoint III). Whenever necessary, we emphasize theparticular viewpoint used, by notationally specifying the lamination L inquestion as symbolic lamination L A , algebraic lamination L , or as laminarylanguage L respectively.For currents, the transition between the three viewpoints is also canonical(see [Kap04]), and we will freely move from one to the other without alwaysnotifying the reader. To be specific, the Kolmogorov value µ ( w ) of a reducedword w = y . . . y k ∈ F ( A ), from the viewpoint III, is precisely the measureof the cylinder C A ( w ) = { . . . z i − z i z i +1 . . . | z = y , . . . , z k = y k } ⊂ Σ A from viewpoint I, and also, corresponding to viewpoint II, equal to the mea-sure of the algebraic cylinder C A ( w ) ⊂ ∂ F N given by { ( X, wX ′ ) | X = x x . . . , X ′ = x ′ x ′ . . . ∈ ∂F ( A ) , x = y , x ′ = y − k } . Note that the algebraic cylinder C A ( w ) is the image of the “symbolic” cylin-der C A ( w ) under the map Σ A → ∂ F N , Z = Z − · Z + ( Z − − , Z + ). Remark 2.1.
The reader should notice that in viewpoints I and III a basis A of F N is crucially used, while II is “algebraic”. It is very important toremember that basis change induces on the Kolmogorov function a more5omplicated operation than just rewriting the given group element in thenew basis B . The correct transition is given, for any reduced word w ∈ F ( B ),by decomposing the algebraic cylinder C B ( w ) ⊂ ∂ F N into a finite disjointunion of translates u i C A ( v i ) of properly chosen algebraic cylinders C A ( v i ),with u i , v i ∈ F ( A ), and posing: µ ( w ) = X i µ ( v i ) . Similarly as for laminations (see § w of F N r { } (or rather, every non-trivial conjugacy class) defines an integer current µ w , given (in the language of viewpoint I) as follows: If w = u m for themaximal exponent m ≥
1, then the measure µ w ( C ) of any measurable set C ⊂ Σ A is equal to m times the number of elements of C ∩ L A ( u ), where L A ( u )is the finite set of biinfinite words of type . . . vv · vv . . . , and v ∈ F ( A ) is any ofthe cyclically reduced words conjugated to u or to u − . Alternatively, (in thelanguage of viewpoint II) the current µ w is given by an F N -equivariant Diracmeasure µ w on ∂ F N , defined as follows: For every measurable set C ⊂ ∂ F N the value of µ w ( C ) is given by the number of cosets g < w > ⊂ F N whichcontain an element v that satisfies v ( w −∞ , w ∞ ) ∈ C or v ( w ∞ , w −∞ ) ∈ C . Athird equivalent definition of µ w (corresponding to viewpoint III) is given bya count of “frequencies”, see [Kap03]. The noteworthy fact that µ w dependsonly on the element w ∈ F N and not on the word w ∈ F ( A ) is obvious inthe second of these definitions, but rather puzzeling if one considers only thefirst or the third.A current is rational if it is a non-negative linear combination of finitelymany integer currents. Remark 2.2.
The above setup of the concept of currents in its various equiv-alent forms, together with the canonical identification F N = F ( A ) for anybasis A of F N , provides the ideal means to see very elegantly that many of theclassical measure theoretic tools from symbolic dynamics do not depend onthe underlying combinatorics of the chosen alphabet, but are rather algebraicin their true nature. Determining the exact point to which ergodic theorytools can be “algebraicized” seems to be a worthy task but goes beyond thescope of this paper. 6 The space Curr ( F N ) The set of currents on F N will be denoted Curr( F N ). It comes naturally withseveral interesting structures, which we will discuss briefly in this section. Wewould like to stress that this space, as well as its projectivization, appears tobe a very interesting and useful tool for many fundamental questions aboutautomorphisms of free groups, and we expect that it will play an importantrole in the future developpement of this subject.First, the set Curr( F N ) of currents carries the weak topology, which forany basis A of F N is induced by the canonical embedding of Curr( F N ) intothe vector space R F ( A ) , given by µ ( µ ( w )) w ∈ F ( A ) . In particular, a familyof currents µ i converges towards a current µ ∈ Curr( F N ) if and only if µ i ( w )converges to µ ( w ) for every w ∈ F ( A ).Next, the same formalism as explained in Remark 2.1 for a basis changedefines canonically an action by homeomorphisms of Out( F N ) on the spaceCurr( F N ), which is formally given, for any α ∈ Aut( F N ) and any µ ∈ Curr( F N ), by α ∗ ( µ )( C ) = µ ( α − ( C )), for every measurable set C ⊂ ∂ F N .This convention defines a left action of Out( F N ): α ∗ ( β ∗ ( µ ))( C ) = β ∗ ( µ )( α − ( C )) = µ ( β − ( α − ( C )))= µ (( αβ ) − ( C )) = ( αβ ) ∗ ( µ )( C )For any integer current µ w , with w ∈ F N r { } , this gives (compare [Kap03,Kap04]): α ∗ ( µ w ) = µ α ( w ) Every current µ defines naturally a lamination L ( µ ) for the free group F N . L ( µ ) can be viewed as an algebraic lamination L ( µ ), i.e. a non-emptysubset of ∂ F N which is closed and invariant under the F N -action and theflip involution, compare [CHL-I]. In this setting, L ( µ ) ⊂ ∂ F N is simply thesupport Supp( µ ) of the Borel measure µ on ∂ F N , i.e. the complement of thebiggest open set (= the union of all open sets) with measure 0. Alternatively, L ( µ ) is given via its laminary language L ( µ ) = { w ∈ F ( A ) | µ ( w ) > } . Werefer to a current µ with support contained in a lamination L simply as aninvariant measure on L . Alternatively, one says that µ is carried by L .A lamination L which has, up to scalar multiples, only one current µ with support L ( µ ) = L , is called uniquely ergodic . The simplest examplesof non-uniquely ergodic laminations are given by the union L of two disjoint7aminations L and L , such that L and L are the support of currents µ and µ respectively (for example rational laminations L = L ( a ) and L = L ( b )for distinct basis elements a, b ∈ A ). For 0 < λ < µ ( λ ) = λµ + (1 − λ ) µ , all with support L . Proposition 3.1.
Recall that we assume N ≥ , and let Λ( F N ) denote thespace of laminations for F N as introduced in [CHL-I]. The mapSupp : Curr ( F N ) → Λ( F N ) , µ L ( µ ) is Out ( F N ) -equivariant, but not continuous and not surjective.Proof. The Out( F N )-equivariance is a direct consequence of the definition ofthe action of Out( F N ) on Curr( F N ) and on Λ( F N ).To see that the map Supp is non-surjective it suffices to consider thesymbolic lamination L = L { a,b } ( Z ) generated by the biinfinite word Z = . . . aaab · aaa . . . . It consists of the σ -orbit of Z and of the periodic word . . . aa · aa . . . , as well as of their inverses. However, it is an easy exercise toshow that any Kolmogorov function µ on the associated laminary language L { a,b } ( Z ), as it takes on values in R ≥ and not in R ≥ ∪ {∞} , must associatethe value 0 to any word that contains the letter b , so that all the measure of µ will be concentrated on the sublamination L { a,b } ( a ) of L .The fact that the map Supp is non-continuous can be seen from theabove defined family µ ( λ ) of currents with constant support L , by lettingthe parameter λ converge inside the open interval (0 ,
1) to the value 0 (or 1):For any such λ the support of µ ( λ ) is clearly the union L ∪ L , while for thelimit one gets L ( µ (0)) = L (or L ( µ (1)) = L ). ⊔⊓ The space Curr( F N ) has some additional structures which are not matchedby corresponding structures in Λ( F N ). For example, there is a canonical lin-ear structure on Curr( F N ), given simply by the embedding of Curr( F N ) intothe real vector space R F ( A ) . Projectivization µ [ µ ] defines the space of projectivized currents P Curr( F N ). Both Curr( F N ) and its projectivization areinfinite dimensional, but P Curr( F N ) is compact. Clearly, the map Supp splitsover the projectivization, thus inducing a map P Supp : P Curr( F N ) → Λ( F N ),which by Proposition 3.1 is Out( F N )-equivariant, non-continuous, and non-surjective. We obtain furthermore 8 emma 3.2. The map P Supp : P Curr ( F N ) → Λ( F N ) is non-injective.Proof. Any non-uniquely ergodic lamination, in particular the above definedfamily µ ( λ ) of currents with constant support L , shows that the map P Suppis not injective. ⊔⊓ A second interesting example for the non-continuity of the map Supp,other than the one given in the proof of Proposition 3.1, is given by therational currents n µ ab n which converge to µ b , while their support L ( ab n )converge to the lamination generated by . . . bba · bb . . . and . . . bb · bb . . . , whichis strictly larger than the lamination L ( b ).This last example, as also the one given in the proof of Proposition 3.1,indicates that a weaker statement than the continuity might be true for themap Supp. Since this will be needed in § δ of Λ( F N ) is saturated if δ contains with anylamination also all of its sublaminations. Lemma 3.3.
Let δ ⊂ Λ( F N ) be a closed saturated subset of laminations.Then the full preimage ∆ ⊂ P Curr ( F N ) of δ under the map P Supp is closed.Proof.
We consider a sequence of currents µ k in Curr( F N ), with L ( µ k ) ∈ δ forany µ k . By the compactness of P Curr( F N ) and of Λ( F N ) we can assume, afterpossibly passing over to a subsequence, that there is a current µ ∈ Curr( F N )and a lamination L ∈ Λ( F N ) with [ µ ] = lim k →∞ [ µ k ] and L = lim k →∞ L ( µ k ). Byproperly normalizing the µ k we can actually assume that µ = lim k →∞ µ k .We now fix a basis A of F N and consider the value of the Kolmogorovfunction µ ( w ) for any w ∈ F N r { } . If µ ( w ) >
0, then by the topology onCurr( F N ), for any ε with µ ( w ) > ε > k such that forany k ≥ k one has | µ k ( w ) − µ ( w ) | < ε . This shows for all k ≥ k that w belongs to the laminary language L ( µ k ). But this implies that w belongs tothe laminary language of L , which shows that µ is carried by L . Since byhypothesis δ is closed and saturated, this shows that [ µ ] is contained in ∆,so that the latter must be closed. ⊔⊓ A weaker statement than the surjectivity of the map Supp is cruciallyused in §
5, again in the proof of Proposition 5.6:9 emma 3.4.
Every lamination L ∈ Λ( F N ) contains a sublamination whichis the support of some current µ ∈ Curr ( F N ) .Proof. For some basis A of F N , let Z = . . . z i − z i z i +1 . . . be a leaf of thelamination L . Let Z n = z − n . . . z n be the central subword of Z of length2 n + 1.For every n ∈ N we define a “counting function” m n : F ( A ) → R ≥ , bysetting, for any word w in F ( A ), m n ( w ) to be the number of occurences of w as subword of Z n or of Z − n , divided by 4 n + 2. It follows directly that m n satisfies the equations that defines the right and the left Kolmogorovproperty, up to possibly an error of absolute value less than n +1 . The totalvalue of m n on the set of words of length 1 is 1, for any n ∈ N . Moreover m n ( w ) is non-zero only for subwords of Z .For each word w in F ( A ) we can chose a subsequence of ( m n ) n ∈ N whosevalue at w converges. By a diagonal argument we get a subsequence thatconverges pointwise to a limit function µ which satisfies the Kolmogorovlaws while still having total value 1 on set of words of length 1, so that it isnon-zero.By construction, we have m n ( w ) = m n ( w − ) for all w ∈ F ( A ), so thatthe same is true for µ . Hence µ is a current. Its support is contained in theset of subwords of Z and thus, as a lamination, in L . ⊔⊓ A very interesting subspace
M ⊂ P Curr( F N ) has been introduced byR. Martin in [Mar95] as closure of the Out( F N )-orbit of [ µ a ], for any element a of any basis A of F N . R. Martin shows that a projectivized integer current[ µ w ] belongs to M if and only if w is contained in a proper free factor of F N .In contrast to the analogous situation for Out( F N ) L ( a ) (compare Proposi-tion 8.1 of [CHL-I]), for N ≥ M is the unique minimal subspace of Curr( F N ) which is non-empty, closedand Out( F N )-invariant.The fact that currents behave somehow more friendly than laminations isunderlined by the following fact, proved in R. Martin’s thesis and attributedthere to M. Bestvina (compare to Proposition 6.5 of [CHL-I]): Proposition 3.5 ([Mar95]) . The set of projectivized integer currents [ µ w ] ,for any w ∈ F N , is dense in P Curr ( F N ) . Geometric currents
A large class of very natural examples for a current µ ∈ Curr( F N ) is givenby any geodesic lamination L ⊂ S , provided with a transverse measure µ ′ ,where S is a hyperbolic surface with boundary as considered in the section 3of [CHL-I] and section 6 of [CHL-II]. In this case the measure µ on ∂ F N canbe nicely seen geometrically through the canonical identification of ∂F N withthe space ∂ e S of ends of the universal covering e S , which is embedded as subsetin the boundary at infinity S ∞ = ∂ H . Two disjoint intervals A, B ⊂ S ∞ ,with intersections A ′ = A ∩ ∂ e S, B ′ = B ∩ ∂ e S , define a measurable set A ′ × B ′ of ∂ F N , and the measure µ ( A ′ × B ′ ) is precisely given by the measure µ ′ ( β )of an arc β in S which is transverse to L , and which lifts to an arc e β in e S ⊂ H that has its two endpoints on the two extremal leaves of e L ⊂ e S which bound the set of all leaves of e L that have one endpoint in A and oneendpoint in B . R -trees In this section we assume familiarity of the reader with the notions of [CHL-II],from which we also import the notation without further explanations.In the last section we have seen that every transverse measure µ on ageodesic lamination L which is contained in a hyperbolic surface S , withnon-empty boundary and with an identification π S = F N , gives rise to acanonical current in Curr( F N ) which we also denote by µ . In section 6 of[CHL-II] we have discussed that ( L , µ ) determines an R -tree T µ with isometric F N -action, and that the support of the current µ and the dual laminationof T µ are the same: this lamination is precisely the lamination associated to L ⊂ S .One of the most intriguing aspects of the relationship between currentsand R -trees comes from the attempt to extend this correspondence, whichfor surfaces is almost tautological, to more general R -trees T . Indeed, thegoal of this section is to understand better the true nature of the interactionbetween the metric on T and an invariant measure µ carried by the duallamination L ( T ) as defined in [CHL-II].In the sequel we consider the dual lamination L ( T ) as algebraic lamination L ( T ), i.e. a non-empty, F N -invariant, flip-invariant and closed subset of11 F N . From [CHL-II] we know that there is a map Q : L ( T ) → T which is F N -equivariant and continuous (see Proposition 8.3 of [CHL-II]). Here T isan element of the boundary ∂ cv N of the unprojectivized Outer space cv N : inparticular, T is a non-trivial R -tree with minimal, very small F N -action byisometries (see [CHL-II], § F N -orbits of points aredense in T (“ T has dense orbits”), and we denote by T the metric completionof T . Corollary 5.1.
For all T ∈ ∂ cv N with dense orbits, the map Q : L ( T ) → T is measurable (with respect to the two Borel σ -algebras on L ( T ) and on T ). ⊔⊓ We apply the last corollary in order to define an extended pseudo-metric d µ on T , for any current µ which is carried by L ( T ). An extended pseudo-metric is just like a metric, except that distinct points P, Q may have distance0, positive distance, or distance ∞ . Definition 5.2.
Let T ∈ ∂cv N be with dense orbits, and assume that µ ∈ Curr( F N ) satisfies Supp( µ ) ⊂ L ( T ). One then defines, for any P, Q ∈ T ,their µ -distance as follows: d µ ( P, Q ) = µ (( Q ) − ([ P, Q ])) [ = Q ∗ ( µ )([ P, Q ]) ]Clearly the function d µ is symmetric and, since T is a tree, it satisfies thetriangular inequality. For three points P, Q, R ∈ T with Q ∈ [ P, R ] one has d µ ( P, R ) = d µ ( P, Q ) + d µ ( Q, R ) unless µ (( Q ) − ( { Q } )) >
0, which of coursecan happen (for example if Q has non-trivial stabilizer which carries all ofthe support of µ ).We distinguish now three special cases (note that we always assume that T is a minimal R -tree, so that it agrees with its interior): The metric d µ iscalled zero throughout T if any two points in T have µ -distance 0. It is called infinite throughout T if any two distinct points in T have µ -distance ∞ . Itis called positive throughout T if any two distinct points in T have positivefinite µ -distance. Otherwise we call the µ -distance mixed .A particular case, which is of special importance, is the following: Definition 5.3. An R -tree T ∈ ∂cv N is called dually uniquely ergodic if thedual lamination L ( T ) is uniquely ergodic.12e note that, in the case where T is dually uniquely ergodic, the µ -distance is uniquely determined by T , up to rescaling. In this case we sup-press the measure µ and speak simply of the dual distance d ∗ on T . Conjecture 5.4. If T is dually uniquely ergodic then the dual distance isnot mixed. We finish this article by proving that the case of dual distances whichare infinite or zero throughout the interior does actually exist, and that itoccurs in a natural context. We assume from now on a certain familiaritywith some of the modern tools for the geometric theory of automorphismsof free groups. Background material and references can be found in [Vog02].In particular we will use below the following facts and definitions:
Remark 5.5. (1) An automorphism α of F N is called irreducible with irre-ducible powers (iwip) if no non-trivial proper free factor of F N is mapped byany positive power of α to a conjugate of itself.(2) It is known (compare [LL03]) that for every iwip automorphism α thereis, up to F N -equivariant homothety, precisely one minimal forward limit R -tree T α ∈ ∂cv N which admits a homothety H : T α → T α with stretchingfactor λ α > α . By this we mean that α ( w ) H = Hw : T α → T α holds for every w ∈ F N . Note that both, the map H as well as the F N -action on T , extend canonically to the metric completion T α , so that the laststatement holds also for T α instead of T α .(3) In terms of the induced action of Out ( F N ) on the non-projectivized closedOuter space cv N (see [CHL-II], § T α ∗ = α − ∗ T = λ α T , where λ α T denotes the tree T rescaled by the factor λ α .(4) As a consequence of the equation in (2), the homothety H satisfies: H Q = Q α : L ( T α ) → T α . (5) There is no further fixed point of the α ∗ -action on CV N other than thepoints [ T α ] and [ T α − ] specified above. In [LL03] it is shown that any iwipautomorphism has North-South dynamics on CV N .136) One knows from [Mar95], Theorem 30 (again attributed to M. Bestvina)that, if α is not geometric , i.e. induced by a surface homeomorphism h : S ≈ → S via some identification F N ∼ = π S , then the α ∗ -action on P Curr( F N )possesses precisely two fixed points, an attractive and a repelling one, andthat α ∗ has a North-South dynamics on P Curr( F N ).(7) Let us denote by µ α ∈ Curr( F N ) a representative of the attracting fixedpoint of the α ∗ -action on P Curr( F N ). It satisfies α ∗ ( µ α ) = λ α µ α , see [Mar95],where λ α is the stretching factor given in (2).(8) Following [Mar95], the support of µ α is contained in the so called legallamination L α ∈ Λ( F N ): Its leaves are represented, for any train track repre-sentative f : τ → τ of α , by biinfinite legal paths in τ , and consequently bynon-trivial (in fact: biinfinite) geodesics in T α (compare with the attractivelamination defined in [BFH97]). In particular, it follows from the alternativedefinition of the dual lamination, L ( T ) = L Q ( T ), given in Theorem 1.1 of[CHL-II], that the two laminations L α and L ( T α ) are disjoint.(9) Any iwip automorphism possesses a train track representative f : τ → τ with transition matrix that is primitive. As a consequence, any edge e of τ will have an iterate f k ( e ) which crosses over all other edges. The canonicalimage in T α (under the map i : e τ → T α , see [LL03]) of any lift of f k ( e ) tothe universal covering e τ is a segment which has the property that the unionof its F N -translates covers all of T α . Proposition 5.6.
For every non-geometric iwip automorphisms α ∈ Aut ( F N ) ,the forward limit tree T α is dually uniquely ergodic.Proof. From the Out( F N )-equivariance of the map λ : ∂cv N → Λ( F N ) inProposition 9.1 of [CHL-II], together with Remark 5.5 (3) above, it fol-lows that the dual lamination L ( T α ) is fixed by α . Hence the set ∆( α ) ⊂ P Curr( F N ), which consists of all preimages under the map P Supp of the lam-ination L ( T α ) and any of its sublaminations, is invariant under the action of α ∗ (by the equivariance of the maps Supp and P Supp, see Proposition 3.1).As the set of all sublaminations of a given lamination is closed, see Proposi-tion 6.4 of [CHL-I], it follows from Lemma 3.3 that ∆( α ) is closed. Further-more ∆( α ) is non-empty, by Lemma 3.4. Thus ∆( α ) is the non-empty unionof closures of α ∗ -orbits, so that it must contain the closure of at least one α ∗ -orbit in P Curr( F N ). From the North-South dynamics of the α ∗ -action on P Curr( F N ) (Remark 5.5 (6)) it follows that either ∆( α ) consists of preciselyone of the two fixed points [ µ α ] or [ µ α − ], or else it contains both of them.14ut according to Remark 5.5 (8) the support of µ α is contained in thelegal lamination L α , which in turn is disjoint from L ( T α ). Hence [ µ α ] isnot contained in ∆( α ), which proves that the latter consists precisely ofthe point [ µ α − ]. This shows that L ( T α ) supports only one (projectivized)current, namely [ µ α − ]. ⊔⊓ We can now give the proof of our main result as stated in § Proof of Theorem 1.1.
From Proposition 5.6 and its proof we know that theforward limit tree T α has dual lamination L ( T α ) which carries an (up tohomothety) unique current, and that this current is equal to µ α − .We now calculate, for any P, Q ∈ T α (using Remark 5.5 (4) to get thethird, and (7) to get the sixth of the equalities below): d ( H ( P ) , H ( Q )) = λ α d ( P, Q )and d ∗ ( H ( P ) , H ( Q )) = µ α − (( Q ) − ([ H ( P ) , H ( Q )]))= µ α − (( H Q α − ) − ([ H ( P ) , H ( Q )]))= µ α − ( α (( Q ) − ([ P, Q ])))= α − ∗ ( µ α − )(( Q ) − ([ P, Q ]))= λ α − µ α − (( Q ) − ([ P, Q ]))= λ α − d ∗ ( P, Q )Assume now that some points P = Q ∈ T α have finite dual distance. Byiterating H one finds an interval [ H n ( P ) , H n ( Q )] with the property that theunion of its F N -translates covers all of T α (compare Remark 5.5 (9)). Thisimplies that any two points in T α have finite dual distance. If the dualdistance function is furthermore non-zero, by the same argument it followsthat any two points have non-zero distance. Thus the dual metric d ∗ on T α defines a non-trivial R -tree T ∗ α with free F N -action, and hence, since theequation in Remark 5.5 (2) carries over from T α to T ∗ α , the R -tree T ∗ α definesa fixed point [ T ∗ α ] of the α ∗ -action on ∂CV N (see § § T ∗ α ] must agree with either [ T α ] or [ T α − ].But this cannot be because we computed above that the streching factor ofthe α ∗ -action on T ∗ α is equal to λ α − and hence bigger than 1 (which rulesout [ T ∗ α ] = [ T α − ]), but different from λ α (thus ruling out [ T ∗ α ] = [ T α ], byRemark 5.5 (3)).Hence the dual metric d ∗ must be either zero or infinite throughout T α . ⊔⊓
15 concrete example of an automorphism that satisfies the propertiesstated in Theorem 1.1 as hypotheses is given in [ABHS05] by the automor-phism a abb acc a of F , which has streching factor 1 , . . . , while its inverse a cb c − ac c − b has streching factor 1 , . . . .An iwip automorphism α ∈ Aut( F N ) is called parageometric , if α is notgeometric, but T α is a geometric tree (see [GL95, GJLL98]). It has beenproved recently in [HM04], see also [Gui04], that in this case the iwip au-tomorphism α − is not parageometric, and that its stretching factor λ α − is strictly smaller than λ α (compare [Gau05]). A family of such automor-phisms, one for any N ≥
3, has been exhibited and investigated in[JL98].We summarize:
Corollary 5.7.
The dual metric on the forward limit tree of any parageo-metric iwip automorphism of F N , or of its inverse, is always infinite or zerothroughout. References [ABHS05] P. Arnoux, V. Berth´e, A. Hilion, and A. Siegel. Fractal repre-sentation of the attractive lamination of an automorphism of thefree group.
Ann. Inst. Fourier (Grenoble) , 2161–2212, 2006[BFH97] M. Bestvina, M. Feighn, and M. Handel. Laminations, trees, andirreducible automorphisms of free groups. Geom. Funct. Anal. ,215–244, 1997.[Bon86] F. Bonahon. Bouts des vari´et´es hyperboliques de dimension 3. Ann. of Math. , 71–158, 1986.16Bon88] F. Bonahon. The geometry of Teichm¨uller space via geodesiccurrents.
Invent. Math. , 139–162, 1988.[CHL05] T. Coulbois, A. Hilion, and M. Lustig. Non-uniquely ergodic R -trees are topologically determined by their algebraic lamination.Preprint, 2005. (To appear in Illinois J. Math.)[CHL-I] T. Coulbois, A. Hilion, and M. Lustig. R -trees and laminationsfor free groups I: Algebraic laminations. ArXiv:math/0609416.[CHL-II] T. Coulbois, A. Hilion, and M. Lustig. R -trees and lamina-tions for free groups II: The lamination associated to an R -tree.ArXiv:math/0702281.[Fur02] A. Furman. Coarse-geometric perspective on negatively curvedmanifolds and groups. In Rigidity in dynamics and geometry(Cambridge, 2000) , 149–166. Springer, Berlin, 2002.[GJLL98] D. Gaboriau, A. Jaeger, G. Levitt, and M. Lustig. An indexfor counting fixed points of automorphisms of free groups.
DukeMath. J. , 425–452, 1998.[GL95] D. Gaboriau and G. Levitt. The rank of actions on R -trees. Ann.Sci. ´Ecole Norm. Sup. , 549–570, 1995.[Gau05] F. Gautero. Combinatorial mapping-torus, branched surfaces andfree group automorphisms. Preprint 05.[Gui04] V. Guirardel. Core and intersection number for group actions ontrees. ArXiv:math/0407206.[HM04] M. Handel and L. Mosher. The expansion factors of an outerautomorphism and its inverse. ArXiv:math/0410015.[JL98] A. J¨ager and M. Lustig. Free Group Automorphisms with ManyFixed Points at Infinity. Preprint 1998[Kap03] I. Kapovich. The frequency space of a free group. Internat. J.Alg. Comput. (special Gaeta Grigorchuk’s 50’s birthday issue), , 939–969, 2005. 17Kap04] I. Kapovich. Currents on free groups. “Topological and Asymp-totic Aspects of Group Theory” (R. Grigorchuk, M. Mihalik, M.Sapir and Z. Sunik, Editors), AMS Contemporary MathematicsSeries , , 149–176, 2006.[KL06] I. Kapovich and M. Lustig, The actions of Out ( F n ) on the bound-ary of Outer space and on the space of currents: minimal sets andequivariant incompatibility. ArXiv:math/0605548. (To appear inErgodic Theory and Dyn. Systems.)[LL03] G. Levitt and M. Lustig. Irreducible automorphisms of F n havenorth-south dynamics on compactified outer space. J. Inst. Math.Jussieu , 59–72, 2003.[Mar95] R. Martin. Non-Uniquely Ergodic Foliations of Thin Type, Mea-sured Currents and Automorphisms of Free Groups . Ph.D.-thesis,UCLA, 1995.[Mor86] J. Morgan. Group actions on trees and the compactification ofthe space of classes of SO(n,1)-representations.
Topology , 1–33, 1986.[Sko96] R. Skora. Splittings of surfaces. J. Amer. Math. Soc. , 605–616,1996.[Vog02] K. Vogtmann. Automorphisms of free groups and outer space. Geom. Dedicata94