The Patterson-Sullivan embedding and minimal volume entropy for outer space

THE PATTERSON-SULLIVAN EMBEDDING AND MINIMAL VOLUME ENTROPY FOR OUTER SPACE ILYA KAPOVICH AND TATIANA NAGNIBEDA Abstract. Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every k≥2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank kand without degree-one vertices is equal to (3k−3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths. 1. Introduction Ageodesic current on a word-hyperbolic group Gis a positive G-invariant Borel measure on the space ∂2G:= {(x, y) : x, y ∈∂G, x 6=y}, where ∂G is the hyperbolic boundary of Gendowed with the canonical boundary topology. The study of geodesic currents on free groups is motivated by investigating geometry and dynamics of individual automorphisms, as well as of groups of automorphisms of a free group. A similar program proved to be successful in the case of fundamental groups of hyperbolic surfaces. Bonahon’s foundational work [4, 5] showed the relevance of geodesic currents to the study of the geometry of the Teichm¨uller space and of the dynamical properties of surface homeomorphisms. Results about geodesic currents in the hyperbolic surface case can be also found in [6, 7, 20, 38, 36] and other sources. Interesting applications of geodesic currents to the study of free group automorphisms were recently obtained in [24, 25, 26, 23]. 2000 Mathematics Subject Classification. Primary 20F65, Secondary 05C, 37A, 37E, 57M. Key words and phrases. free groups, metric graphs, Patterson-Sullivan measures, geodesic currents, volume entropy. The first author was supported by the NSF grant DMS#0404991. Both authors acknowledge the support of the Centre de Recerca Matem`atica at Barcelona and of the Swiss National Foundation for Scientific Research. 1 2 ILYA KAPOVICH AND TATIANA NAGNIBEDA Patterson-Sullivan measures were introduced by Patterson [34] and Sullivan [41] in the context of a Kleinian group acting on the boundary of a hyperbolic space. The notion was extended by Coornaert [10] to the case of a group Gacting geometrically (that is isometrically, properly discontinuously and cocompactly) on a Gromov-hyperbolic geodesic metric space. Let us remind here the definition in the case of a non-elementary group G acting geometrically on a CAT (−1) space X. For s > 0 an s-conformal density is a G-equivariant family of regular Borel measures (µx)x∈Xon ∂X that are pairwise absolutely continuous and with the property that their mutual Radon-Nykodim derivatives satisfy dµx dµy (ξ) = e−sBξ(x,y),for every x, y ∈X, where for a point ξ∈∂X and for x, y ∈X,Bξ(x, y) is a Busemann function defined by Bξ(x, y) := lim z→ξ,z∈X[d(x, z)−d(y, z)]. It turns out that there is a unique s > 0 such that a nonzero s-conformal density exists, namely the critical exponent h(X) of X(see Section 3 below for definition). Moreover, when Gacts on Xgeometrically, there is a unique nonzero h(X)-conformal density up to scalar multiplication. Any of these proportional families (µx)x∈Xis said to be a family of Patterson-Sullivan measures on ∂X. In this setting h(X), which is defined in terms of the Poincar´e series, is equal to the Hausdorff dimension of ∂X. Furman [16] proved that in this case (as well as in the more general situation of a geometric action on a Gromov-hyperbolic space) there is a unique up to scalar multiple nonzero G-invariant measure νon ∂2X:= {(x, y)|x, y ∈∂X, x 6=y}in the same measure class as µ2 x, where (µx)x∈Xis a family of Patterson-Sullivan measures on ∂X. Any of the nonzero scalar multiples of νis called an X-Patterson-Sullivan current. Via the identification between ∂2Gand ∂2X, this measure νpulls back to a canonical, up to a scalar multiple, geodesic current on G, any nonzero scalar multiple of which is called a G-Patterson-Sullivan current. In the case of closed hyperbolic surfaces Patterson-Sullivan currents admit several other equivalent characterizations. Let Sbe a closed surface with a fixed hyperbolic metric ρ, so that ^ (S, ρ) = H2. Thus G=π1(S) acts on H2geometrically and H2/G =S. In this situation there is a natural identification between the space of G-invariant measures on ∂2H2and the space of shift-invariant measures on the unit tangent bundle US, where the R-shift action is given by the geodesic flow on (S, ρ). As shown by PATTERSON-SULLIVAN EMBEDDING 3 Kaimanovich [22], under this identification Patterson-Sullivan currents correspond precisely to Bowen-Margulis measures (or maximal entropy measures) on US, that is the only shift-invariant measures on USwhose entropy is equal to the topological entropy of the geodesic flow on (S, ρ). For this reason Furman suggests, even in the general setting of a group acting geometrically on a Gromov-hyperbolic space, to call Patterson-Sullivan currents Bowen-Margulis or Patterson-Sullivan-Bowen-Margulis currents. For closed hyperbolic surfaces H2-Patterson-Sullivan currents coincide with Liouville currents corresponding to the hyperbolic structure ρ. Bonahon [4, 5] proved that the map sending a marked hyperbolic structure to the corresponding projective class of Liouville currents provides a topological embedding L:T(S)→PCurr(G) of the Teichm¨uller space T(S) to the projectivized space of geodesic currents PCurr(G). The space PCurr(G) is compact and, as was also proved by Bonahon, the closure of L(T(S)) turns out to be homeomorphic to the Thurston compactification [ T(S) of T(S). The Culler-Vogtmann outer space [13] is a free group analogue of the Teichm¨uller space. For a free group Fof finite rank k≥2 the outer space CV (F) consists of equivalence classes of free, discrete and minimal isometric actions of Fon R-trees for which the quotient metric graph has volume one. Two such actions are equivalent if there is an F-equivariant isometry between the two trees in question. Let Fbe a free group of finite rank k≥2, let Γ be a finite connected graph with no degree-one and degree-two vertices, and let α:F→π1(Γ, p) be an isomorphism. Every choice Lof a volume-one metric graph structure on Γ (that is, assignment of positive lengths to non-oriented edges of Γ, so that the sum of the lengths of all edges is equal to 1) defines an action of Fvia αon the R-tree e Γ, and hence defines a point in CV (F). Varying the lengths of edges of Γ gives an open simplex Wαin CV (F) of dimension N−1, where Nis the number of non-oriented edges of Γ. Thus the outer space CV (F) is a union of open simplices of arbitrary large dimension. There is a natural map τ:CV (F)→PCurr(F) that takes a point of CV (F) represented by the action of Fon a tree X, to the projective class of F-Patterson-Sullivan currents corresponding to this action. We call τthe Patterson-Sullivan map or the Patterson-Sullivan-Bowen-Margulis map. Our main result is the following statement, which parallels the above mentioned theorem of Bonahon for hyperbolic surfaces: Theorem A. The Patterson-Sullivan map τ:CV (F)→PCurr(F)is a topological embedding. The Hausdorff dimension map h:CV (F)→R is continuous and, moreover, the restriction of hto any open simplex in CV (F)is real-analytic. 4 ILYA KAPOVICH AND TATIANA NAGNIBEDA Injectivity of τfollows from a general result of Furman [16] proved in the context of geometric actions on Gromov-hyperbolic spaces. The main work in the present paper is in proving the continuity of τ. Any isomorphism α:F→π1(Γ, p) provides a sort of “coordinate system” for PCurr(F). We give an essentially explicit computation of τ|Wαin this coordinate system. The continuity of hcan also be derived by a direct argument using the volume entropy interpretation of hgiven in Lemma 3.9 below. The key point is that if Land L0are two “nearby” metric structures on a finite graph Γ defining the metrics dand d0on X=e Γ then the identity map Id : (X, d)→(X, d0) is bi-Lipschitz with a bi-Lipshitz constant close to 1. As we mentioned earlier, in the case of a closed hyperbolic surface Swith G=π1(S) Bonahon proved that the Liouville map L:T(S)→PCurr(G) extends to a homeomorphism from Thurston’s compactification [ T(S) of T(S) to the closure of the image of L. It is well-known that [ T(S) coincides with the length-function compactification of T(S). We expect that, unlike the map Lin the surface case, the map τ:CV (F)→PCurr(F) does not extend to a homeomorphism from the length function compactification of CV (F) to the closure of τ(CV (F)) in PCurr(F). We intend to pursue this nontrivial question in future work. Reiner Martin [30] constructed a different family of continuous Out(F)-equivariant embeddings from CV (F) to PCurr(F). Unlike the Patterson-Sullivan embedding τ, Martin’s embeddings are not based on a natural geometric construction and use an ad-hoc procedure, where a point of CV (F) is sent to an explicitly defined infinite linear combination of “counting” currents determined by conjugacy classes of elements of F. It is well-understood that in a fairly general negatively curved setting the Hausdorff dimension of the boundary coincides with the volume entropy. If (M, g) is a closed connected Riemannian manifold, then the volume entropy of gis defined as h(g) := lim inf R→∞ log V ol˜g(B(x, R)) R, where x∈f Mis a base-point and B(x, R) is the ball of radius Rand center xin f M, equipped with the pullback ˜gof the Riemannian metric g. This definition does not depend on the choice of x∈f M, and h(g)>0 if and only if the group π1(M) has exponential growth. If ghas strictly negative sectional curvature, then (f M, ˜g) is a CAT(−1) space and the Hausdorff dimension of its boundary (which is also equal to the critical exponent of (f M, ˜g)) is equal to the volume entropy h(g). A similar statement is true for the universal cover of a compact locally CAT(−1)-space K, as stated in PATTERSON-SULLIVAN EMBEDDING 5 Lemma 3.9 below. In that case volume has to be interpreted as counting the number of π1(K)-orbit points in the ball of radius naround the basepoint in e K. For a compact connected Riemannian manifold Mit is natural to ask what the infimum of h(g) is when gvaries over metrics with V olg(M) = 1 and whether this infimum is achieved. This is known as the minimal entropy problem (see discussion in [1]). A famous theorem of Besson, Courtois and Gallot [1] shows that if Madmits a locally symmetric volume-one metric g0 of negative curvature, then g0minimizes volume entropy among all volumeone metrics (see an earlier paper of Katok [27] for the case of surfaces.) A particular case of their theorem (see [2], Section 5) says that if (M, g0) and (M0, g) are homotopically equivalent negatively curved compact connected Riemannian manifolds of the same dimension n≥3, and if (M, g0) is locally symmetric, then hn(g)V ol(M0, g)≥hn(g0)V ol(M, g0). Besson, Courtois and Gallot also show that h(g)=h(g0) and V ol(M0, g)=V ol(M, g0) if and only if (M0, g) is isometric to (M, g0). In the last section of our paper we prove an analogue of the first of these two statements in the outer space setting. Theorem A implies that the volume entropy function h(which again coincides with the Hausdorff dimension of the boundary and with the critical exponent) factors to a continuous function on the moduli space M=CV (F)/Out(F) h:M → R>0. A point in Mis a finite connected graph Γ without degree-one and degreetwo vertices and with π1(Γ) ∼ =F, endowed with the structure Lof a volumeone metric graph. Then h(L) is the volume entropy of the tree ∂e Γ, where the metric on e Γ is given by the lift of L. The analogue of a locally symmetric manifold is a regular graph (i.e., such that all vertices are of the same degree) with all edges of equal length. The volume entropy of a regular tree with all edges of the same length is easy to compute explicitly. In particular, assigning the length 1/(3k−3) to each of (3k−3) non-oriented edges in a trivalent graph with the fundamental group free of rank kgives a volume-one metric graph with volume entropy of its universal cover equal to (3k−3) log 2. We prove that this is precisely the minimum of the volume entropy over all finite connected metric volume-one graphs without vertices of degree one or two and with fundamental group free of rank k. Theorem B. For the function h:CV (F)/Out(F)→Rwe have min h= (3k−3) log 2. 6 ILYA KAPOVICH AND TATIANA NAGNIBEDA This minimum is realized by any regular trivalent connected graph Γwith π1(Γ) ∼ =F, (so that Γhas 3k−3non-oriented edges), where each edge of Γ is given length 1/(3k−3). Moreover, sup M h=∞. As an intermediate step in proving Theorem B we establish that among all the volume-one metric structures on an m-regular graph Γ with m≥ 3, the volume entropy is minimized by assigning all the edges of Γ equal lengths. Gilles Robert [37] had earlier proved this fact under an extra assumption that Γ has a highly transitive automorphism group, by different methods. The authors are grateful to Florent Balacheff, David Berg, Pierre de la Harpe, Vadim Kaimanovich, J´erˆome Los, Martin Lustig, Paul Schupp and Dylan Thurston for helpful discussions. 2. Geodesic currents Convention 2.1. For the remainder of the paper let Fbe a finitely generated free group of rank k≥2. We will denote by ∂F the space of ends of Fwith the standard ends-space topology. Thus ∂F is a topological space homeomorphic to the Cantor set. We shall also think about ∂F as the hyperbolic boundary of F, endowed with the canonical boundary topology, in the sense of the theory of word-hyperbolic groups (see, for example [17]). We will also denote ∂2F:= {(ζ, ξ) : ζ, ξ ∈∂F and ζ6=ξ}. Definition 2.2 (Geodesic currents).Ageodesic current on Fis a positive locally finite (that is, finite on compact subsets) F-invariant Borel measure on ∂2F. We denote the space of all geodesic currents on Fby Curr(F). The space Curr(F) comes equipped with a weak topology: for νn, ν ∈ Curr(F) we have lim n→∞ νn=νiff for every two disjoint closed-open sets S, S0⊆∂F we have lim n→∞ νn(S×S0) = ν(S×S0). Definition 2.3 (Projectivized geodesic currents).For two nonzero geodesic currents ν1, ν2∈Curr(F) we say that ν1is equivalent to ν2, denoted ν1∼ ν2, if there exists a nonzero scalar r∈Rsuch that ν2=rν1. We denote PCurr(F) := {ν∈Curr(F) : ν6= 0}/∼ and call it the space of projectivized geodesic currents on F. The space PCurr(F) is endowed with the quotient topology. We will denote the ∼- equivalence class of a nonzero geodesic current νby [ν]. PATTERSON-SULLIVAN EMBEDDING 7 Convention 2.4. For a (finite or infinite) graph ∆, denote by V∆ the set of all vertices of ∆, and denote by E∆ the set of all oriented edges of ∆ (i.e., the set of all ordered pairs (u, v) where uand vare adjacent vertices in ∆.) A path γin ∆ is a sequence of oriented edges which connects a vertex o(γ) (origin) with a vertex t(γ) (terminus). A path is called reduced if it does not contain a back-tracking (a path of the form (e, e−1)). We denote by P(∆) the set of all finite reduced paths in ∆. For a vertex x∈V∆, we denote by Px(∆) the collection of all γ∈ P(∆) that begin with x. For γ∈ P(∆), we denote by a(γ) the set of all e∈E∆ such that eγ ∈ P(∆) and we denote by b(γ) the set of all e∈E∆ such that γe ∈ P(∆). Definition 2.5 (Simplicial charts).Let Fbe a free group of rank k≥2 and let Γ be a finite connected graph without degree-one vertices such that π1(Γ) ∼ =F. Let α:F→π1(Γ, p) be an isomorphism, where pis a vertex of Γ. We call such αasimplicial chart for F. Convention 2.6. Let α:F→π1(Γ, p) be a simplicial chart. We consider X:= e Γ, a topological tree, and denote the covering map from Xto Γ by j:X→Γ. For γ∈ P(X) we call the reduced path j(γ) in Γ the label of γ. As there is only one reduced path connecting two arbitrary vertices in a tree, we will often write [x, y] for a path in Xwith origin xand terminus y. Let ∂X denote the space of ends of Xwith the natural ends-space topology. Then we obtain a canonical α-equivariant homeomorphism ˆα:∂F → ∂X, as follows. Suppose we endow Γ with the structure of a metric graph, that is, we assign a positive length to each edge of Γ. This turns Xinto an R-tree with a discrete isometric action of π1(Γ, p). Moreover, Xis quasiisometric to Fand, if Fis equipped with a word metric and x0is a lift of pto Xthen the orbit map ˜α:F→X,f→α(f)x0, is a quasi-isometry. This quasi-isometry extends to a homeomorphism ˆα:∂F →∂X. A crucial feature of this construction is that ˆαdoes not depend on the choice of a metric structure on Γ. If αis fixed, we will usually suppress explicit mention of ˆαand also of the map αitself when talking about the action of Fon Xand on ∂X arising from this situation. We also denote by ∂2Xthe set of all pairs (ζ1, ζ2) such that ζ1, ζ2∈∂X and ζ16=ζ2. For (ζ1, ζ2)∈∂2X we denote by [ζ1, ζ2] the simplicial (non-parameterized) geodesic from ζ1 to ζ2in X. Thus [ζ1, ζ2] is a subgraph of Xisomorphic to the simplicial line, together with a choice of direction on that line. We also have the identification ˆα:∂2F→∂2X. Definition 2.7 (Cylinder sets).For every reduced path γin Xdenote 8 ILYA KAPOVICH AND TATIANA NAGNIBEDA CylX(γ) := {(ζ1, ζ2)∈∂2X:γ⊆[ζ1, ζ2] and the orientations on γand on [ζ1, ζ2] agree} Also, for x=o(γ)∈Xdenote Cylx(γ) := {ζ∈∂X :γis an initial segment of [x, ζ]} The collection of all sets CylX(γ), where γvaries over P(X), gives a basis of closed-open sets for ∂2X. Similarly, for any x∈X, the collection of all sets Cylx(γ), where γvaries over Px(X), gives a basis of closed-open sets for ∂X. Let us denote Cylα(γ) := ˆα−1CylX(γ), so that Cylα(γ)⊆∂2F. It is easy to see that: Lemma 2.8. For νn, ν ∈Curr(F) lim n→∞ νn=νiff lim n→∞ νn(Cylα(γ)) = ν(Cylα(γ)) for every γ∈ P(X). Moreover, for ν, ν0∈Curr(F)we have ν=ν0iff ν(Cylα(γ)) = ν0(Cylα(γ)) for every γ∈ P(X). Remark 2.9. Note that for any f∈Fand γ∈ P(X) we have fCylα(γ) = Cylα(fγ). Since geodesic currents are, by definition, F-invariant, for a geodesic current νand for γ∈ P(X) the value ν(Cylα(γ)) only depends on the label j(γ) of γ. 3. Patterson-Sullivan measures We recall here the basics regarding Patterson-Sullivan measures in the context of CAT(−1) spaces. As already mentioned in the Introduction, the original theory of Patterson-Sullivan measures for Kleinian groups was developed by Patterson [34] and Sullivan [41]. This theory was generalized to groups acting on CAT (−1) spaces by Burger and Mozes [9], Coornaert- Papadopoulos [11, 12] and other authors (see, for example [21, 35]). The general case of Gromov-hyperbolic spaces was given careful treatment by Coornaert [10]. Later Furman [16], also working in the general Gromovhyperbolic context, obtained length spectrum rigidity results and explained the connection between Patterson-Sullivan measures and the analogues of Bowen-Margulis currents. The Patterson-Sullivan measures on the universal covers of finite simplicial graphs were considered by Lyons [29] and by Coornaert and Papadopoulos [11]. Although all of the above-mentioned articles are relevant to our work, the primary references for us are Burger-Mozes[9], Coornaert-Papadopoulos[11], Coornaert [10] and Furman [16]. The work of Furman is particularly important in our context since there Patterson-Sullivan currents (as opposed to Patterson-Sullivan measures or conformal densities) are treated in great PATTERSON-SULLIVAN EMBEDDING 9 generality. Moreover, the injectivity of the Patterson-Sullivan map in Theorem A is an immediate corollary of Furman’s results [16]. Definition 3.1 (Metric and semi-metric graph structures).Asemi-metric graph structure Lon a (finite or infinite) graph Γ is an assignment of the length L(e)≥0 to each edge e∈EΓ of Γ in such a way that L(e) = L(e−1) for every e∈EΓ. We say that such a structure Lis non-singular if there is a maximal tree Tin Γ such that L(e)>0 for every e∈E(Γ−T). We define the volume of Las vol(L) := 1 2Pe∈EΓL(e). Thus vol(L) can be thought of as the sum of the lengths of the non-oriented edges of Γ. A semi-metric graph structure Lis called a metric graph structure if L(e)>0 for each e∈EΓ. If Lis a semi-metric graph structure on Γ, let Γ0be the graph obtained from Γ by contracting to points all edges of Γ of L-length zero. Then Γ0comes equipped with a canonical metric graph structure L0coming from L. We call (Γ0,L0)the metric graph associated to (Γ,L). Convention 3.2. Let Lbe a nonsingular semi-metric graph structure on a finite graph Γ. Let (Γ0,L0) be the metric graph associated to (Γ,L) and let q: Γ →Γ0be the canonical projection map. Let X=e Γ and let j:X→Γ be the covering map. Then Lcanonically lifts to a semi-metric graph structure e Lon Xdefined as e L(e) := L(j(e)) for every e∈EX. Similarly let X0=e Γ0and let j0:X0→Γ0be the associated covering map. Again, L0lifts to a metric graph structure e L0on X0. It is easy to see that both jand j0preserve edge-lengths and that X0is obtained from Xby contracting all edges of length zero in Xto points. Thus (X0,e L0) is the metric graph associated to (X, e L). We denote by eq:X→X0 the canonical projection map. The semi-metric structure e Ldefines a semi-metric d=dLon Xand e L0 defines a metric d0=d0 Lon X0. Moreover, eq: (X, d)→(X0, d0) is distancepreserving. Note that for both (X, d) and (X0, d0) there are obvious notions of geodesic edge-paths. In both cases we can metrize ∂X and ∂X0by setting dx(ξ, ζ) := e−d(x,[ξ,ζ]) where ξ, ζ ∈∂X d0 x0(ξ0, ζ0) := e−d0(x0,[ξ0,ζ0]) where ξ0, ζ0∈∂X0, where x∈X, x0∈X0. Note that dxis a metric on ∂X, although Lwas just a semi-metric structure. Moreover, if x0=eq(x) then the map eq: (∂X, dx)→ (∂X0, d0 x0) is a homeomorphism and an isometry. Convention 3.3. For the remainder of this section we will fix G, X and the notations below to be one of the following: 16 ILYA KAPOVICH AND TATIANA NAGNIBEDA mij := (1,if eiej∈ P(Γ) 0,otherwise. Denote AL(s) := Diag(e−sL(e1), . . . , e−sL(en))M. The system (*) from part (2) of Proposition 4.3 rewrites as the matrix equation: AL(s)   we1 . . . wen   =   we1 . . . wen   . Let ΦL(s) denote the spectral radius of AL(s). Lemma 5.3. The following hold: (1) The matrices AL(s)and AL(s)Tare nonnegative and irreducible for every s∈R. (2) The function ΦL(s)is continuous and strictly monotone decreasing on the interval 0≤s < ∞. (3) We have ΦL(0) >1. Proof. Recall that Γ is finite, connected, has no degree-one vertices and π1(Γ) is a free group of rank k≥2. Therefore the graph H(Γ) is strongly connected and hence its adjacency matrix Mis nonnegative irreducible and the same is true for its transpose MT. The matrix AL(s) is obtained from Mby multiplying the i-th row of Mby a positive number e−sL(ei)for each i= 1, . . . , n. Hence AL(s) and AL(s)Tare nonnegative and irreducible. The continuity of ΦL(s) follows from its definition. Suppose now that 0 ≤s<s0. Let Y=   y1 . . . yn   be a positive Perron- Frobenius eigenvector of AL(s), so that AL(s)Y= ΦL(s)Y. Since L(ei)≥ 0, the functions e−sL(ei)are monotone non-increasing for each i. Hence component-wise aij(s)≤aij(s0) and therefore AL(s0)Y≤AL(s)Y=ΦL(s)Y. Moreover, there is some edge eiwith L(ei)>0 and hence [AL(s0)Y]i< [AL(s)Y]i= ΦL(s)yi. Therefore ΦL(s0)<ΦL(s), as claimed. Finally, note that AL(0) = Mand ΦL(0) is the Perron-Frobenius eigenvalue of M. The fundamental group of H(Γ) is free of rank at least two. Hence the universal cover of H(Γ) has exponential growth, that is, the spectral radius of Mis bigger than 1. ¤ Corollary 5.4. For every non-singular semi-metric structure Lon Γthere exists a unique s > 0such that ΦL(s) = 1, namely s=hL(X). PATTERSON-SULLIVAN EMBEDDING 17 Proof. Lemma 5.3 implies that there is at most one s > 0 such that ΦL(s) = 1. The existence of Patterson-Sullivan measures(Proposition-Definition3.12) and Proposition 4.3 guarantee that when s=hL(X), the Perron-Frobenius eigenvalue of AL(s) is equal to 1, that is, that ΦL(hL(X))) = 1. ¤ We will now rewrite the system (*) AL(s)Y=Yin the form allowing to apply the Implicit Function Theorem. This system is equivalent to the following nequations: e−sL(ei)(mi1y1+... +minyn)−yi= 0, i = 1, . . . , n. To express y1, . . . , yn, s as implicit functions of L(e1), . . . , L(en) we need an extra normalizing equation: y2 1+· · · +y2 n= 1. Proposition 5.5. Let L1=L(e1), . . . , Ln=L(en)be a non-singular semimetric structure Lon Γ. Put Fi(L1, . . . , Ln, y1, . . . , yn, s) := e−sLi(mi1y1+... +minyn)−yi for i= 1, . . . , n, and Fn+1(L1, . . . , Ln, y1, . . . , yn, s) := y2 1+· · · +y2 n−1. Consider the following system of n+ 1 equations in 2n+ 1 variables: (!) Fi(L1, . . . , Ln, y1, . . . , yn, s) = 0, i = 1, . . . , n + 1 Let Jbe the Jacobian of this system, that is the (n+1)×(n+1)-matrix consisting of the partial derivatives of F1, . . . , Fn+1 with respect to y1, . . . , yn, s: Jij =(∂Fi ∂yj1≤i≤n+ 1,1≤j≤n ∂Fi ∂s 1≤i≤n+ 1, j =n+ 1. Suppose s > 0, yi>0, for i= 1, . . . , n, are such that the point z= (L1, . . . , Ln, y1, . . . , yn, s)satisfies the system (!). Then det J|z6= 0. Proof. Let us compute the matrix Jat z, using the information that z satisfies (!). We will denote aij = (AL(s))ij =e−sLimij. For i6=j, 1 ≤i, j ≤nwe get ∂Fi ∂yj=e−sLimij =aij . For i=jwe get ∂Fi ∂yi=e−sLimii −1 = aii −1. Thus in the upper left corner of Jwe see the n×nmatrix AL(s)−In. Let us compute ∂Fi ∂s . We have ∂Fi ∂s =−Lie−sLi(mi1y1+... +minyn) = −Liyifor i= 1, . . . , n, where the last equality holds since Fi(z) = 0. Finally, the last row of Jobtained by differentiating Fn+1 =y2 1+···+ y2 n−1 along y1, .., yn, s is [2y12y2. . . 2yn0]. 18 ILYA KAPOVICH AND TATIANA NAGNIBEDA Thus J=             a11 −1a12 a13 . . . a1n−L1y1 a21 a22 −1a23 . . . a2n−L2y2 . . .. . .. . .. . . . . .. . . ai1ai2ai3. . . ain −Liyi . . .. . .. . .. . . . . .. . . an1an2an3. . . ann −1−Lnyn 2y12y22y3. . . 2yn0             We claim that the rows of the matrix Jare linearly independent and hence det J6= 0. The column vector Y=   y1 . . . yn   satisfies (AL(s)−In)Y= 0. This implies that the last row of Jis perpendicular to the first nrows. Since Y > 0, it therefore suffices to show that the first nrows of Jare linearly independent. Note that det(AL(s)−In) = 0. However the matrix AL(s)−Inhas rank n−1 since 1 is the Perron-Frobenius eigenvalue of AL(s) and hence has multiplicity one. Thus, up to a scalar, there is only one nontrivial linear relation between the rows of AL(s)−In. This relation is given by the left Perron-Frobenius eigenvector Q= [q1, . . . , qn] of AL(s). Indeed QAL(s) = Qand Q[AL(s)−In] = 0. Note that qi>0 for all i= 1, . . . , n. Suppose that the first nrows of Jare linearly dependent and that we have a nonzero row vector Zof length nsuch that ZJn= 0 where Jnis the n×(n+ 1) matrix consisting of the first nrows of J. Then Zis also a relation between the first nrows of AL(s)−Inand hence Zis a multiple of Q. Thus QJn= 0. However, when we multiply Qby the last column of Jnto compute the (n+ 1)-st entry in QJn, we get −L1y1q1−L2y2q2− · · · − Lnynqn. This number is strictly negative since Li≥0, yi>0, qi>0 for all i= 1, . . . , n and there is some isuch that Li>0. This gives us a contradiction with the fact that QJn= 0. ¤ For the remainder of this section we will denote an n-tuple (p1, . . . , pn)∈ Rnby p. Corollary 5.6. Let L(0) 1=L(0)(e1), . . . , L(0) n=L(0)(en)be a non-singular semi-metric structure L(0) on Γ. Suppose s(0) >0, y(0) i>0, where i= 1, . . . , n, are such that the point z(0) = (L(0), y(0), s(0))∈R2n+1 satisfies the system (!). Then there exist an open neighborhood Uof L(0) in Rnand PATTERSON-SULLIVAN EMBEDDING 19 real-analytic functions s=s(L),yi=yi(L)on Usuch that for every L∈U the point (L, y1(L), . . . , yn(L), s(L)) ∈R2n+1 satisfies (!) and such that yi(L(0)) = y(0) i,s(L(0)) = s(0). Moreover, whenever L∈Udefines a non-singular semi-metric structure Lon Γ, then s(L)is equal to the critical exponent of (e Γ, dL)and (y1(L), . . . , yn(L)) is a scalar multiple of (we1,L, . . . , wen,L). Proof. Proposition 5.5 implies that the Implicit Function Theorem is applicable at the point z(0). Thus there exists an open neighborhood Uof z(0) and real-analytic functions s=s(L), yi=yi(L) on Usuch that for every L∈Uthe point (L, y1(L), . . . , yn(L), s(L)) ∈R2n+1 satisfies (!) and such that yi(L(0)) = y(0) i,s(L(0)) = s(0). Moreover, since y(0) i>0, we can choose Uso that yi=yi(L)>0 on U. Let L∈Udefine a non-singular semi-metric structure Lon Γ. By Proposition 4.3 the critical exponent h=hLof (e Γ, dL) satisfies the property that ΦL(h) = 1. Also, by construction, ΦL(s(L)) = 1. Corollary 5.4 now implies that h=s(L). Moreover, Proposition 4.3 and the definition of the functions yi(L1, . . . , Ln) imply that both (we1,L, . . . , wen,L) and (y1(L), . . . , yn(L)) are Perron-Frobenius eigenvectors of the matrix AL(h). Therefore they are scalar multiples of each other, as required. ¤ 6. Patterson-Sullivan currents The following is essentially a corollary of Proposition 1 of Furman [16]. Proposition-Definition 6.1 (Patterson-Sullivan current).Let G, X be as in Convention 3.3. Let (µx)x∈Xbe a family of Patterson-Sullivan measures on ∂X and let µ=µyfor some y∈X. Then there exists a unique, up to a scalar multiple, G-invariant and flip-invariant nonzero locally finite measure νon ∂2Xin the measure class of µ×µ. Moreover, this measure νis of the form dν(ξ, ζ) = e2hfµ(ξ,ζ)dµ(ξ)dµ(ζ), 20 ILYA KAPOVICH AND TATIANA NAGNIBEDA where fµ:∂2X→R+is a symmetric Borel function which is within bounded distance from the function d(x, [ξ, ζ]), and his the critical exponent of X. Such a measure νis called an X-Patterson-Sullivan current for this action of Gon X. Since νis unique up to a scalar multiple, its projective class [ν] is called the projective X-Patterson-Sullivan current. For the remaining part of this section let F, Γ,L, X, d, α be as in Part 2) of Convention 3.3. Definition 6.2. Recall that the choice of simplicial chart αdefines a homeomorphism ˆα:∂2F→∂2X. Let νbe an X-Patterson-Sullivan current. Then its pull-back ˆα∗(ν) is an F-invariant measure on ∂2Fwhich is called an F-Patterson-Sullivan current for the pair (α, L). Its projective class [ν] is called the projective F-Patterson-Sullivan current for the pair (α, L). We now proceed to give an explicit formula for the X-Patterson-Sullivan current associated with the action of Fon X. Proposition 6.3. Let z∈X, and let h=hL(X)be the critical exponent of X. Let (µx)xbe a family of Patterson-Sullivan measures on ∂X and let webe defined as in Convention 4.2. Then the measure νon ∂2Xgiven by the formula (♣)dν(ξ, ζ) = e2hd(z,[ξ,ζ])dµz(ξ)dµz(ζ) is an X-Patterson-Sullivan current. Moreover, for any path γ= [x, y]∈ P(X)with label j(γ)∈ P(Γ) we have (†)ν(CylX(γ)) = e−hL(γ)¡X e∈b(e0) we¢¡ X e∈b(e00) we¢ where (e0)−1∈EΓis the label of the first edge of γand e00 ∈EΓis the label of the last edge of γ. Proof. We will first show that (†) defines a geodesic current on ∂X. That is, we claim that there exists a unique geodesic current ν0such that for every γas in the statement of the proposition ν0(CylX(γ)) = e−hL(γ)¡X e∈b(e0) we¢¡ X e∈b(e00) we¢. In view of the definition of we’s the above formula is equivalent to (‡)ν0(CylX(γ)) = e−hL(γ)µx(Cyly(γ−1))µy(Cylx(γ)). PATTERSON-SULLIVAN EMBEDDING 21 The uniqueness of ν0is obvious. Also, by construction ν0is F-invariant, provided that ν0is a measure. Thus it remains to show that the above formula does define a measure on ∂2X. To do this we need to check (see, for example, [25]) that for every γas above ν0(CylX(γ)) = X f∈b(γ) ν0(CylX(γf)) and ν0(CylX(γ)) = X f∈a(γ) ν0(CylX(fγ)). We will verify the first formula, as the second one is completely analogous. By (‡) applied to each of the paths γf, where f∈b(γ), we have ν0(CylX(γf)) = e−hL(γf)µx(Cylt(f)(f−1γ−1))µt(f)(Cylx(γf)) = e−hL(γf)µx(Cyly(γ−1))µt(f)(Cylx(γf)) = e−hL(γf)µx(Cyly(γ−1))µt(f)(Cyly(f)). Since Cylx(γ) = G f∈b(γ) Cylx(γf), it follows that µy(Cylx(γ)) = X f∈b(γ) µy(Cylx(γf)). Therefore ν0(CylX(γ)) = e−hL(γ)µx(Cyly(γ−1))µy(Cylx(γ)) = e−hL(γ)µx(Cyly(γ−1))¡X f∈b(γ) µy(Cylx(γf))¢= e−hL(γ)µx(Cyly(γ−1))¡X f∈b(γ) µy(Cyly(f))¢= by Remark 4.1 e−hL(γ)µx(Cyly(γ−1))¡X f∈b(γ) e−hL(f)µt(f)(Cyly(f))¢= X f∈b(γ) e−hL(γf)µx(Cyly(γ−1))µt(f)(Cyly(f)) = X f∈b(γ) ν0(CylX(γf)). 22 ILYA KAPOVICH AND TATIANA NAGNIBEDA Thus ν0is indeed a geodesic current. We will now show that ν0=ν, where the measure νon ∂2Xis defined by (♣). It suffices to show that ν(CylX(γ)) = ν0(CylX(γ)) for every γ∈ P(X). Let γ= [x, y]∈ P(X). We need to consider the following two cases. Case 1: d(z, [x, y]) >0. Let z0∈[x, y] be such that d(z, z0) = d(z, [x, y]). Then ν(CylX(γ)) = e2hd(z,z0)µz(Cylz([z, x]))µz(Cylz([z, y])) = e2hd(z,z0)e−hd(z,z0)µz0(Cylz0([z0, x]))e−hd(z,z0)µz0(Cylz0([z0, y])) = µz0(Cylz0([z0, x]))µz0(Cylz0([z0, y])) = e−hd(z0,x)µx(Cylz0([z0, x]))e−hd(z0,y)µy(Cylz0([z0, y])) = e−hd(x,y)µx(Cylz0([z0, x]))µy(Cylz0([z0, y])) = e−hd(x,y)µx(Cyly([y, x]))µy(Cylx([x, y])) = e−hL(γ)µx(Cyly(γ−1))µy(Cylx(γ)) = ν0(CylX(γ)). Case 2: d(z, [x, y]) = 0 (so that z∈[x, y]). We will assume that z6=x, z 6=y. The argument is easily adapted for the cases z=xor z=y. Then ν(CylX(γ)) = e2hd(z,[x,y])µz(Cylz([z, x]))µz(Cylz([z, y])) = µz(Cylz([z, x]))µz(Cylz([z, y])) = e−hd(z,x)µx(Cylz([z, x]))e−hd(z,y)µy(Cylz([z, y])) = e−hd(x,y)µx(Cylz([z, x]))µy(Cylz([z, y])) = e−hd(x,y)µx(Cyly([y, x]))µy(Cylx([x, y])) = e−hL(γ)µx(Cyly(γ−1))µy(Cylx(γ)) = ν0(CylX(γ)). Therefore ν=ν0, which completes the proof of Proposition 6.3 ¤ 7. The Culler-Vogtmann outer space The Culler-Vogtmann outer space, introduced by Culler and Vogtmann in a seminal paper [13], is a free group analogue of the Teichm¨uller space of a closed surface of negative Euler characteristic. We refer the reader to [13, 8, 3, 19] for a detailed discussion and the proofs of the basic facts that are listed in this section. Definition 7.1 (Outer space).Let Fbe a free group of finite rank k≥2. A marked metric graph structure on Fis a pair (α, L), where α:F→π1(Γ, p) is a simplicial chart for Fand Lis a metric graph structure on Γ. A marked PATTERSON-SULLIVAN EMBEDDING 23 metric graph structure is minimal if Γ has no degree-one and degree-two vertices. Two marked metric graph structures (α1:F→π1(Γ1, p1),L1) and (α2: F→π1(Γ2, p2),L2) are equivalent if there exist an isometry ι: (Γ1,L1)→ (Γ2,L2) and a path vfrom ι(p1) to p2in Γ2such that (ι#◦α1)(f) = vα2(f)v−1 for every f∈F. Clearly, minimality is preserved by equivalence of marked metric graph structures. The Culler-Vogtmann outer space CV (F) consists of equivalence classes of all volume-one minimal marked metric graph structures on F. Definition 7.2 (Elementary charts).Let α:F→π1(Γ, p) be a simplicial chart for F, where Γ has no degree-one and degree-two vertices. For each non-singular semi-metric structure Lon Γ let Γ0,L0and qbe as in Convention 3.2. Then q#◦α:F→π1(Γ0, q(p)) is a simplicial chart for Fand (q#◦α, L0) is a minimal marked metric graph structure on F. Denote by S(Γ) the set of all volume-one non-singular semi-metric structures on Γ. Note that if Γ has Nnon-oriented edges, then S(Γ) is embedded as a subset of Rn. We topologize S(Γ) accordingly. It is not hard to see that for two non-singular semi-metric structures L1,L2on Γ the pairs (q#◦α, L0 1) and (q#◦α, L0 2) are equivalent if and only if L1=L2. Thus αdefines an injective map λα:S(Γ) →CV (F), λα:L 7→ (q#◦α, L0). This map λαis called the elementary chart in CV (F) corresponding to α. Let now S+(Γ) denote the set of all metric structures on Γ. If Γ has n oriented edges then S+(Γ) is an open simplex of dimension n/2−1 in Rn and S+(Γ) is dense in S(Γ). Definition 7.3 (Topology on the outer space).The outer space CV (F) is endowed with the weakest topology for which every elementary chart is a topological embedding. As explained in [13], the outer space CV (F) is a union of open simplices of the form λα(S+(Γ)), where λαis as in Definition 7.2. One can also view CV (F) as the space of projectivized hyperbolic length functions on F corresponding to free and discrete isometric actions of Fon R-trees. Definition 7.4 (Projectivized length functions).Let FLen(F) denote the space of all hyperbolic length functions `:F→Ron Fcorresponding to free and discrete isometric actions of Fon R-trees. The space FLen(F) is endowed with the weak topology of pointwise convergence. We will say that two length functions in F Len(F) are equivalent if they are scalar multiples of each other, and will denote by PFLen(F) the space 24 ILYA KAPOVICH AND TATIANA NAGNIBEDA of equivalence classes of elements of FLen(F), endowed with the quotient topology. The equivalence class of `∈FLen(F) is denoted [`]. For each `∈FLen(F) there exists a free discrete minimal isometric action of Fon an R-tree X`such that `is the hyperbolic length function for this action. Moreover, the tree X`and the corresponding action of Fare unique up to an equivariant isometry. Let Γ`denote the metric graph X`/F. Let FLen1(F) denote the set of all `∈FLen(F) such that Γ`has volume one. Note that every equivalence class [`]∈PFLen(F) has a unique representative in FLen1(F). For each `∈FLen(F) the action of Fon X`defines an isomorphism α`:F→π1(Γ`, p), where p∈VΓ`. Let L` denote the metric graph structure on Γ`inherited from X`. Note that the equivalence class of the marked metric graph structure (α`,L`) on Fdoes not depend on the choice of p. The following result is well-known [13]: Proposition 7.5. (1) The restriction of the quotient map [ ] : FLen(F)→PFLen(F)to FLen1(F)is a homeomorphism whose image is PFLen(F). Thus F Len1(F)is canonically homeomorphic to PFLen(F). (2) Let %:FLen1(F)→CV (F)be the map that takes each `∈ FLen1(F)to the equivalence class of the marked structure (α`,L`) on F. Then %:F Len1(F)→CV (F)is a homeomorphism. Thus the outer space CV (F) is homeomorphic to the spaces F Len1(F) and PFLen(F). 8. Proof of the main result If (α1:F→π1(Γ1, p1),L1) and (α2:F→π1(Γ2, p2),L2) are two equivalent pairs representing the same point η∈CV (F), then R-trees X1=e Γ1 and X2=e Γ2are F-equivariantly isometric and the corresponding hyperbolic length functions are equal. Hence it follows from Proposition 2 of Furman [16] (and it is also easy to see this directly) that the projective F- Patterson-Sullivan currents corresponding to (α1,L1) and (α2,L2) coincide (see Definition 6.2). Hence the following map is well-defined: Definition 8.1 (Patterson-Sullivan map and Hausdorff dimension map). Let Fbe a free group of finite rank k≥2 and let CV (F) denote the outer space. Let η∈CV (F). Thus ηis represented as an equivalence class of (α, L), where α:F→π1(Γ, p) is a simplicial chart on Fsuch that Γ is a finite connected graph without degree-one and degree-two vertices and where L is a volume-one metric graph structure on Γ. Let X=e Γ and let dbe the PATTERSON-SULLIVAN EMBEDDING 25 metric on Xinduced by L. Define τ(η) to be the projective F-Patterson- Sullivan current on Fcorresponding to (α, L). Also define h(η) to be the Hausdorff dimension of ∂X (which, as we have seen, is equal to the critical exponent hL(X).) This defines a map τ:CV (F)→PCurr(F), which we will call the Patterson-Sullivan map, and a map h:CV (F)→R, which we will call the Hausdorff dimension map. Theorem 8.2. The Patterson-Sullivan map τ:CV (F)→PCurr(F)is a topological embedding. The Hausdorff dimension map h:CV (F)→R is continuous and, moreover, the restriction of hto any open simplex in CV (F)is real-analytic. Proof. Since CV (F) is locally compact, in order to prove that τis a topological embedding it suffices to show that τis continuous and injective. Injectivity follows from Theorem 2 of Furman [16]. Recall the identification of CV (F) with PF Len(F) from Proposition 7.5. If τ([`1]) = τ([`2]) for `1, `2∈FLen(F) then Theorem 2 of [16] implies that there is r > 0 such that r`1=`2and hence [`1] = [`2]. We now establish that τand hare continuous. Since every point of the outer space is contained in finitely many elementary charts, it suffices to prove that τand hare continuous on the image of every elementary chart in CV (F). Let α:F→π1(Γ, p) be a simplicial chart for F, where Γ has no degreeone and degree-two vertices. Let λαbe the elementary chart in CV (F) determined by α. Recall that the image Im(λα) of λαconsists of all points of CV (F) corresponding to volume-one semi-metric structures on Γ where all the edges with zero length are contained in a (possibly empty) subtree of Γ. Corollary 5.6 and formula (†) in Proposition 6.3 imply that τ|Im(λα)and h|Im(λα)are continuous and, moreover, the restriction of hto the interior of Im(λα) is real-analytic. ¤ Remark 8.3 (Out(F)-Equivariance).It is easy to see that the Patterson- Sullivan map τis equivariant with respect to the left action of Out(F) and, in fact, a similar statement holds in the general word-hyperbolic context considered by Furman [16]. It is even easier to see that his constant on each Out(F)-orbit and thus factors to a continuous map on the moduli space h:CV (F)/Out(F)→R. Indeed, suppose (α:F→π1(Γ,p),L) represents a point η∈CV (F) and let φ∈Aut(F). Let X=e Γ, equipped with the metric dinduced by L. By definition of the left action of Aut(F) (and of Out(F)) on CV (F), the point φη ∈CV (F) is the equivalence class of (φ−1◦α, L). For both 32 ILYA KAPOVICH AND TATIANA NAGNIBEDA [22] V. Kaimanovich, Bowen-Margulis and Patterson measures on negatively curved compact manifolds. Dynamical systems and related topics (Nagoya, 1990), 223– 232, Adv. Ser. Dynam. Systems, 9, World Sci. Publishing, River Edge, NJ, 1991 [23] V. Kaimanovich, I. Kapovich and P. Schupp, The Subadditive Ergodic Theorem and generic stretching factors of group homomorphisms, in preparation [24] I. Kapovich, The frequency space of a free group, to appear in Internat. J. Alg. Comput.; http://www.arxiv.org/math.GR/0311053 [25] I. Kapovich, Currents on free groups, preprint, 2004; http://www.arxiv.org/math.GR/0412128 [26] I. Kapovich, G. Levitt, P. Schupp and V. Shpilrain, Translation equivalence in free groups, Transact. Amer. Math. Soc., to appear; http://www.arxiv.org/math.GR/0409284 [27] A. Katok, Entropy and closed geodesics. Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 339–365 [28] B. Kitchens, Symbolic dynamics. One-sided, two-sided and countable state Markov shifts. Universitext. Springer-Verlag, Berlin, 1998 [29] R. Lyons, Equivalence of boundary measures on covering trees of finite graphs, Ergodic Theory Dynam. Systems 14 (1994), no. 3, 575–597 [30] R. Martin, Non-Uniquely Ergodic Foliations of Thin Type, Measured Currents and Automorphisms of Free Groups, PhD Thesis, UCLA, 1995 [31] R. D. Mauldin and S. Williams, Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829 [32] T. Nagnibeda, Random walks on trees and Ramanujan graphs. in: “Random walks and Geometry”, ed. V. Kaimanovich, de Gruyter, 2004, pp. 487–500 [33] T. Nagnibeda, W. Woess, Random walks on trees with finitely many cone types. J. Theoret. Probab. 15 (2002), pp. 383-422 [34] S. Patterson, Lectures on measures on limit sets of Kleinian groups. Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), 281–323, London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, Cambridge, 1987 [35] F. Paulin, On the critical exponent of a discrete group of hyperbolic isometries. Differential Geom. Appl. 7(1997), no. 3, 231–236 [36] J.-C. Picaud, Cohomologie bornee des surfaces et courants geodesiques. Bull. Soc. Math. France 125 (1997), no. 1, 115–142 [37] G. Robert Entropie et graphes. Pr´epublication 182, ENS Lyon, 1996 [38] D. Sari´c, Infinitesimal Liouville distributions for Teichm¨uller space. Proc. London Math. Soc. (3) 88 (2004), no. 2, 436–454 [39] D. Sari´c, Geodesic currents and Teichm¨uller space, Topology 44 (2005), no. 1, 99–130 [40] E. Seneta, Nonnegative matrices and Markov chains. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1981 [41] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes ´ Etudes Sci. Publ. Math. 50 (1979), 171–202 PATTERSON-SULLIVAN EMBEDDING 33 Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA http://www.math.uiuc.edu/~kapovich/ E-mail address:[email protected] Section de math´ematiques, Universit´e de Gen`eve, 2-4, rue du Li`evre, c.p.64, 1211 Gen`eve, Switzerland http://www.math.kth.se/~tatiana/ E-mail address:[email protected]