Automorphisms of graphs of cyclic splittings of free groups
AAutomorphisms of graphs of cyclic splittings of free groups
Camille Horbez and Richard D. WadeMay 12, 2019
Abstract
We prove that any isometry of the graph of cyclic splittings of a finitely generatedfree group F N of rank N ≥ is induced by an outer automorphism of F N . The samestatement also applies to the graphs of maximally-cyclic splittings, and of very smallsplittings. Introduction
The study of the outer automorphism group of a finitely generated free group has benefitedgreatly from analogies with mapping class groups of compact surfaces. In recent years,research has concentrated on understanding the geometry of proposed analogues of the curvegraph of a surface, one main question being that of finding natural hyperbolic Out ( F N ) -graphs. Among these stand the free factor graph, the free splitting graph and the cyclicsplitting graph. We refer the reader to [5, 6, 7, 14, 15, 16, 17, 19, 22] for various resultsabout these graphs.In the context of closed orientable surfaces, Royden proved that (except in a finitenumber of sporadic cases) the group of biholomorphisms of Teichmüller space, as well asits group of isometries with respect to the Teichmüller metric, coincides with the extendedmapping class group of the surface [25]. His result was extended by Earle and Kra tothe case of punctured surfaces [12]. Similar results are also known to hold for the Weil–Petersson metric [23, 9] or Thurston’s asymmetric metric [30]. Ivanov, Korkmaz and Luo’srigidity result for the curve graph of a (nonsporadic) compact surface [18, 20, 21] states thatthe group of simplicial isometries of this graph also coincides with the extended mappingclass group. The reader is referred to [24] for a list of various rigidity results for simplicialactions of mapping class groups.There are some known analogous results for the group Out ( F N ) . Bridson and Vogt-mann [8] first proved that when N ≥ , the group Out ( F N ) is the group of simplicialautomorphisms of the spine of Outer space, and Francaviglia and Martino then used thisto show that the group of isometries of Outer space with the Lipschitz metric [13] is alsoequal to Out ( F N ) when N ≥ , and to P SL (2 , Z ) when N = 2 (another approach to this1 a r X i v : . [ m a t h . G R ] J a n esult, based on a study of the metric completion of Outer space, is due to Algom-Kfir [1]).Building on Bridson and Vogtmann’s result, Aramayona and Souto proved that for N ≥ ,the group Out ( F N ) is also the group of simplicial isometries of the free splitting graph [2].This paper is concerned with a rigidity question concerning the isometry group of Mann’scyclic splitting graph [22], and a pair of its close relatives.A splitting of F N is a simplicial tree on which F N acts by simplicial automorphisms withno proper invariant subtree. Two splittings are equivalent if there exists an F N -equivarianthomeomorphism between them. Given two splittings T and T (cid:48) of F N , we say that T isa refinement of T (cid:48) if T (cid:48) is obtained by equivariantly collapsing some of the edges in T to points. Let F Z N denote the graph of cyclic splittings of F N , i.e. the graph whosevertices are the equivalence classes of splittings of F N whose edge stabilizers are cyclic(possibly trivial) subgroups of F N . Two such splittings are joined by an edge if one is aproper refinement of the other. Let F Z maxN be the graph of maximally-cyclic splittings of F N , which is defined in the same way with the extra assumption that edge stabilizers inthe splittings we consider are closed under taking roots. We will also consider the graph V S N of very small splittings of F N . This consists of maximally-cyclic splittings that areadditionally required to have trivial tripod stabilizers. The graphs F Z N , F Z maxN and
V S N ,equipped with the path metric, come with right isometric actions of the group Out ( F N ) ofouter automorphisms of F N (an automorphism φ ∈ Aut ( F n ) induces an isometry of eachcomplex by precomposing each F N –action on a tree by φ , and this isometry depends onlyon the outer automorphism class of φ ). The goal of this paper is to show the followingrigidity result, which states that every isometry of either F Z N , F Z maxN or V S N is in factinduced by the action of an element of Out ( F N ) . Theorem A.
For all N ≥ , the natural maps from Out ( F N ) to the isometry groups of F Z N , F Z maxN and
V S N are isomorphisms. A description of all maximally-cyclic splittings of F can be found in [11]. The complex F Z max turns out to be isomorphic to the Farey graph with depth two dead ends and "fins"attached. In particular, its automorphism group is isomorphic to P SL (2 , Z ) .We may then assume that N ≥ . Let G be one of the graphs F Z N , F Z maxN or V S N .The free splitting graph F S N , whose vertices are the equivalence classes of splittings of F N with trivial edge stabilizers, sits as a subcomplex inside G . Our proof of Theorem Arelies on Aramayona and Souto’s rigidity statement [2] for the free splitting graph: we firstprove that any isometry of G preserves the subgraph F S N setwise, and then show that anyisometry of G which restricts to the identity on F S N is actually the identity map.The restriction to the set of very small splittings is a natural one – these are exactly thecyclic splittings which arise in the boundary of Outer space (see [10] where the notion ofa very small splitting was introduced for the first time). The natural inclusions of F Z maxN and
V S N into F S N may not be quasi-isometries, however Mann’s proof [22] translates toshow that F Z maxN and
V S N are also hyperbolic.2e finally note that the problem of determining the group of simplicial isometries ofthe free factor graph is still open. Acknowledgments
This work started during the programme "The Geometry of Outer space: Investigatedthrough its analogy with Teichmueller space" held at Aix-Marseille Université during Sum-mer 2013. We are greatly indebted to the organizers of this event. We would also like tothank Brian Mann for inspiring conversations we had there. F N Recall that a splitting of F N is a simplicial action of F N on a simplicial tree T with noproper and nontrivial invariant subtrees. A splitting is cyclic if every edge stabilizer is cyclic(possibly trivial), and is maximally-cyclic if each nontrivial edge stabilizer is a maximally-cyclic subgroup of F N (i.e. edge stabilizers are closed under taking roots). When G is one ofthe graphs F S N , F Z N , F Z maxN or V S N defined in the introduction, we say that a splittingis G -maximal if it admits no nontrivial refinement in G .Given an edge e in an F N -tree T adjacent to a vertex v , we denote by [ e ] the G v -orbitof the edge e and by [ G e ] the G v -conjugacy class of its edge group. The following lemmais a version of a theorem by Shenitzer and Swarup [27, 29], see also [28] or [4, Lemma 4.1].This will turn out to be crucial in our proof of Theorem A for understanding the structureof cyclic splittings of F N . Lemma 1.1. (Shenitzer [27], Swarup [29], Stallings [28], Bestvina–Feighn [4, Lemma 4.1])Let T be a cyclic splitting of F N with a nontrivial edge stabilizer. Then there exists an edge e with nontrivial stabilizer G e adjacent to a vertex v such that: ( (cid:63) ) There is a decomposition G v = G e ∗ A such that if e (cid:48) is another edge adjacent to v , with [ e (cid:48) ] (cid:54) = [ e ] , then some representative of [ G e (cid:48) ] is contained in A . We say that an edge e satisfying the condition ( (cid:63) ) is unfoldable at the vertex v . Theabove lemma tells us that we can find a (possibly trivial) refinement T (cid:48) of T by (equivari-antly) replacing the vertex v with the tree corresponding to the above splitting of G v . Inthe new quotient graph T (cid:48) /F N , the vertex with stabilizer G e will have valence 2, and wehave the following lemma: Lemma 1.2. If T is a maximally-cyclic splitting of F N with some nontrivial edge stabilizerthen there is a refinement T (cid:48) of T such that the quotient graph of groups T (cid:48) /F N has a vertexof valence two where one adjacent edge has a trivial stabilizer, and the other is nontrivial.In particular, if T is G -maximal, then in the quotient graph T /F N each unfoldable edge as a vertex of valence two such that the other adjacent edge at this vertex has a trivialstabilizer. Let v be a vertex of valence in T (cid:48) /F N given by Lemma 1.2, and e (resp. e ) be theedge adjacent to v with trivial (resp. nontrivial) stabilizer. We can equivariantly collapsethe edge e in T (cid:48) to obtain a new splitting T (cid:48)(cid:48) . We say that T (cid:48)(cid:48) is obtained from T by unfolding the edge e , and T (cid:48) is a partial unfolding . In the opposite direction T (cid:48) is obtainedfrom T (cid:48)(cid:48) by partially folding the orbit of the edge added to split G v , and T is obtainedfrom T (cid:48)(cid:48) by fully folding this orbit. Note that the unfolding operation is, in general, farfrom unique; there may be many possible choices for the complementary free factor A . Forexample, all splittings of F = (cid:104) a, b, c (cid:105) of the form (cid:104) a, b (cid:105) ∗ (cid:104) c [ a, b ] k (cid:105) with k ∈ Z are obtainedby (fully) unfolding the splitting (cid:104) a, b (cid:105) ∗ [ a,b ] (cid:104) c, [ a, b ] (cid:105) . The structure of one-edge cyclic splittings of F N . A splitting T is a k -edge splitting if there are k orbits of edges in T under the action of F N . To illustrate Lemma 1.1, we givethe following classification of one-edge cyclic splittings of F N . Such a splitting has one ofthe following forms: • a separating one-edge free splitting F N = A ∗ B , where A and B are complementaryproper free factors of F N , or • a nonseparating one-edge free splitting F N = C ∗ , where C is a corank one free factorof F N , or • a separating one-edge Z -splitting F N = A ∗ (cid:104) w (cid:105) ( B ∗ (cid:104) w (cid:105) ) , where A and B are comple-mentary proper free factors of F N , and w ∈ A , or • a nonseparating one-edge Z -splitting F N = ( C ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) , where C is a corank onefree factor of F N , and w ∈ C , and t denotes the stable letter. Two useful results.
We say that two splittings are compatible if they both can be ob-tained by equivariantly collapsing edges of a common tree. Scott and Swarup showed thata k -edge splitting is determined by its set of k one-edge collapses. Furthermore, it is enoughfor a set of splittings to be pariwise compatible to find a common refinement of the wholecollection. Theorem 1.3. (Scott–Swarup [26, Theorem 2.5], see also Handel–Mosher [15, Lemma1.3]) Any set T , . . . , T k of distinct, pairwise-compatible, one-edge cyclic splittings of F N has a unique k -edge refinement. Any k -edge cyclic splitting of F N refines exactly k distinctone-edge splittings. At times we will also need to know the existence of a uniform bound on the number ofedges in maximally-cyclic splittings of F N . This was shown by Bestvina and Feighn.4 heorem 1.4. (Bestvina–Feighn [3]) There is a uniform bound (depending only on N ) forthe number of orbits of edges in a maximally-cyclic splitting of F N . Notice however that there is no bound on the number of orbits of edges in an arbitrarycyclic splitting of F N . For example, the group F = (cid:104) a, b (cid:105) has arbitrarily long splittings ofthe form F = (cid:104) a (cid:105) ∗ (cid:104) a (cid:105) (cid:104) a (cid:105) ∗ (cid:104) a (cid:105) (cid:104) a (cid:105) ∗ (cid:104) a (cid:105) · · · ∗ (cid:104) b (cid:105) . G -maximal splittings Let N ≥ . Throughout the section, we denote by G one of the graphs F S N , F Z N , F Z maxN or V S N . The goal of this section is to provide a characterization of one-edge and G -maximalsplittings in terms of their combinatorics in G . Lemma 2.1.
All G -maximal splittings have finite valence in G .Proof. A G -maximal splitting T is not properly refined by any splitting in G , and thenumber of splittings it properly refines is equal to the number of proper subsets of orbitsof edges in T (see Theorem 1.3), which is finite. Lemma 2.2.
All one-edge cyclic splittings of F N are compatible with infinitely manymaximally-cyclic splittings of F N . In particular, all one-edge splittings have infinite va-lence in G .Proof. Let T be a one-edge cyclic splitting of F N . If T is a free splitting, then as N ≥ , oneof the vertex groups of T has rank at least , and splitting this vertex group yields infinitelymany distinct proper refinements of T (in particular T is compatible with infinitely manyfree splittings).Suppose that T is a separating one-edge Z -splitting of the form F N = A ∗ (cid:104) w (cid:105) ( B ∗ (cid:104) w (cid:105) ) ,where A and B are complementary proper free factors of F N , and w ∈ A . Let { b , . . . , b k } be a free basis of B . Then the splittings A ∗ (cid:104) w (cid:105) (cid:104) w (cid:105) ∗ (cid:104) b w i , b , . . . , b k (cid:105) , where i varies in N ,yield infinitely many distinct two-edge refinements of T (in particular T is compatible withinfinitely many free splittings).Finally, assume that T is a nonseparating one-edge Z -splitting of the form F N = ( C ∗(cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) , where C is a corank one free factor of F N , and w ∈ C , and t denotes the stableletter. Then for each g ∈ C which is not a proper power, the splitting F N = ( C ∗ (cid:104) g t − (cid:105) ) ∗ (cid:104) g (cid:105) is compatible with T , and this yields infinitely many distinct two-edge refinements of T (inparticular T is compatible with infinitely many maximally-cyclic splittings). Lemma 2.3.
For all T ∈ G , the following conditions are equivalent. • The splitting T is either a one-edge splitting or a G -maximal splitting. • There exist splittings T and T of F N such that d G ( T, T ) = d G ( T, T ) = 1 , and – d G ( T , T ) = 2 , and – T − T − T is the unique path of length joining T to T in G .Proof. If T is a G -maximal splitting, then any F N -equivariant partition of the set of edgesof T into two subsets E and E gives rise to distinct splittings T i , obtained by equivariantlycollapsing edges in E i to points, that satisfy the desired properties by Theorem 1.3. If T is aone-edge splitting in G , Lemma 2.2 shows that it is compatible with infinitely many distinctone-edge maximally-cyclic splittings of F N . If all these splittings were pairwise compatible,Theorem 1.3 would enable us to construct maximally-cyclic splittings of F N with arbitrarilylarge numbers of orbits of edges. This would contradict Theorem 1.4. Therefore, we canfind a pair of two-edge proper refinements T and T of T which are not compatible. Suchtrees satisfy the desired properties.Conversely, assume that there exist splittings T and T satisfying the conclusions ofthe lemma. • If T is properly refined by T and T is properly refined by T , then T is properlyrefined by T , so d G ( T , T ) = 1 , a contradiction. • If T and T are both properly refined by T , then T is G -maximal, otherwise we couldfind a proper refinement T (cid:48) of T in G , and get two paths of length joining T to T in G (namely, the path going through T and the path going through T (cid:48) ). • If T is properly refined by both T and T , then T is a one-edge splitting, otherwisewe could find a splitting T (cid:48) in G that is properly refined by T , and get two differentpaths of length joining T to T as above.As d G ( T, T ) = d G ( T, T ) = 1 , up to exchanging T and T one of the above cases occurs,and the claim follows.As a consequence of Lemmas 2.1, 2.2 and 2.3, we get the following result. Proposition 2.4.
Any isometry of G preserves the sets of one-edge splittings and of G -maximal splittings (setwise).Remark . A free splitting of F N is maximal among free splittings of F N if and only ifit is maximal among cyclic splittings of F N (such a splitting has trivial vertex stabilizers,and hence cannot be refined by a splitting having nontrivial edge stabilizers). Hence wecan talk about "maximal free splittings" without any ambiguity. Remark . As the number of edges of
T /F N is equal to the number of one-edge splittingsadjacent to T (Theorem 1.3), any isometry of G preserves the number of edge orbits in eachsplitting. As a consequence, the property that T (cid:48) is obtained from T by either collapsingor adding a fixed number of edges is preserved under an isometry of G .6 Invariance of the free splitting graph
The first step in our proof of Theorem A is to show that an isometry of the graph
F Z N , F Z maxN or V S N preserves the subgraph F S N of free splittings of F N setwise. We do thisby distinguishing maximal free splittings from maximal splittings which have at least onenontrivial edge stabilizer. We first look at the graph F Z N . In this situation the argumentis much simpler as all maximal splittings are free. F Z N . Proposition 3.1.
All
F Z N -maximal splittings are free splittings.Proof. Assume towards a contradiction that there exists a cyclic splitting T of F N thatis F Z N -maximal, and has an edge with nontrivial stabilizer. Lemma 1.2 implies that T contains a vertex v with nontrivial stabilizer G v , which projects to a vertex of valence in the quotient graph T /F N , and is adjacent to an edge e with trivial stabilizer in T . Bytaking a proper power of a generator, we can find g ∈ G v that generates a proper subgroupof G v , and partially fold e along ge in an equivariant way, to obtain a proper refinement of T . This contradicts F Z N -maximality of T . Lemma 3.2.
A splitting of F N is a free splitting if and only if it is at distance at most onefrom a maximal free splitting in F Z N .Proof. A splitting having a nontrivial edge stabilizer cannot be refined by a free splitting.We need to show that any free splitting enlarges to a maximal one. This follows from theexistence of a bound on the number of edges in a free splitting of F N (it is well knownthat maximal free splittings have N − edges but an abstract bound also follows fromTheorem 1.4). Corollary 3.3.
Any isometry of
F Z N preserves the subgraph F S N (setwise).Proof. Propositions 2.4 and 3.1 imply that any isometry of
F Z N preserves the set of max-imal free splittings of F N . The claim then follows from Lemma 3.2.Proposition 3.1 is no longer true for the graphs F Z maxN and
V S N , so in these cases weneed a more refined argument to distinguish maximal free splittings from other G -maximalsplittings. For example, the splitting displayed on Figure 1 is both V S N -maximal and F Z maxN -maximal. However, it fails to be
F Z N -maximal (one can add an extra cyclic edgewith stabilizer ( cd ) into the middle of the splitting).7 b h c, d ih c ih d ih cd i Figure 1: A
V S -maximal tree with nontrivial edge stabilizers. V S N . We saw in Section 2 that
V S N -maximal splittings have finite valence in V S N . We nowdistinguish maximal free splittings from other V S N -maximal splittings, by using the fol-lowing property: if a maximal splitting is free then collapsing one orbit of edges still gives asplitting of finite valence. In contrast, this is not satisfied by V S N -maximal splittings thatcontain an edge with nontrivial stabilizer. Proposition 3.4.
Let T be a maximal free splitting of F N . Any splitting obtained byequivariantly collapsing one edge in T has finite valence in F Z maxN (hence also in
V S N ).Proof. It is a standard fact that all vertices in T have valence , and T contains N − orbits of edges. Equivariantly collapsing an edge that maps to a loop-edge in the quotientgraph of groups creates a tree T (cid:48) whose quotient graph of groups has a vertex of valence with Z as its stabilizer. Any proper refinement of T (cid:48) must be obtained by inserting anedge attached to this vertex, and as we cannot split cyclic vertices along proper powers in F Z maxN , one deduces that the tree T is the only proper refinement of T (cid:48) . Equivariantlycollapsing an edge that does not map to a loop-edge creates a free splitting with trivialvertex groups where one vertex has valence . There are exactly three ways of attachingback an edge to the quotient graph which are given by the possible pairings of edges at thevertex of valence . Hence T (cid:48) has finite valence. Proposition 3.5.
Let T be a V S N -maximal splitting that has some nontrivial edge stabi-lizer. Then there exists an edge e in T such that equivariantly collapsing e to a point yieldsa splitting of infinite valence in V S N .Proof. As T is V S N -maximal, Lemma 1.2 shows the existence of an edge e in T adjacentto a vertex v of valence (in the quotient graph T /F N ) such that its second adjacent edge e (cid:48) has a trivial stabilizer. Let v (cid:48) be the other vertex of e .If the stabilizer of v (cid:48) has rank at least , let T (cid:48) be the splitting obtained by equivariantlycollapsing e to a point. Then T (cid:48) contains a vertex v (cid:48)(cid:48) (the image of v (cid:48) under the collapse8ap) whose stabilizer G v (cid:48)(cid:48) has rank at least , and which is adjacent to an edge e (cid:48) withtrivial stabilizer. Hence T (cid:48) has infinite valence in V S N , since one gets infinitely manydistinct refinements of T (cid:48) by equivariantly folding an initial segment of e (cid:48) with an initialsegment of ge (cid:48) , for some g ∈ G v (cid:48)(cid:48) which is not a proper power (and which can be chosen soas not to create any tripod stabilizer).If the stabilizer of v (cid:48) is cyclic, minimality of the F N -action, together with the fact thatedge stabilizers are not proper powers, prevents v (cid:48) from having valence in the quotientgraph T /F N . As G v (cid:48) is cyclic, edge stabilizers are not proper powers, and e is not containedin a tripod stabilizer, one of the adjacent edges of v (cid:48) has a trivial stabilizer. Let T (cid:48) be thetree obtained from T by equivariantly collapsing e to a point. Then T (cid:48) contains a vertex v (cid:48)(cid:48) with nontrivial cyclic stabilizer (a generator of which we denote by t ), adjacent to twoedges (denoted by e and e ) with trivial stabilizers that have initial segments which arein distinct F N -orbits. Hence T (cid:48) has infinite valence in V S N , since one gets infinitely manydistinct refinements T (cid:48) k of T (cid:48) by equivariantly folding an initial segment of e with an initialsegment of t k e , for k varying in Z . To check that the splittings T (cid:48) k are pairwise distinct, let g ∈ F N be an element whose axis in T (cid:48) crosses the turn ( e , t k e ) (this exists by minimalityof the F N -action on T (cid:48) ), and let h ∈ F N be an element whose axis crosses the turn ( te , t e ) .Then for all l ∈ N , the axes of g and h meet in a single point in T (cid:48) l , except precisely when k = l , in which case they are disjoint. Corollary 3.6.
Any isometry of
V S N preserves the subgraph F S N (setwise).Proof. Let f be an isometry of V S N . Proposition 2.4 shows that f preserves the set of V S N -maximal splittings. Every one-edge collapse of a maximal free splitting has finitevalence (Proposition 3.4), whereas every non-free V S N -maximal splitting has a one-edgecollapse with infinite valence (Proposition 3.5), therefore f preserves the set of maximal freesplittings (Remark 2.6 tells us that the property of being a one-edge collapse of an adjacentvertex is preserved under an isometry). As free splittings are characterized by being atdistance at most one from a maximal free splitting (Lemma 3.2), the map f preserves F S N .Proposition 3.5 fails to be true for the graph F Z maxN , as illustrated by the
F Z max -splitting displayed in Figure 2. Again we will have to refine the argument to distinguishmaximal free splittings from other F Z maxN -maximal splittings.
F Z maxN . We will need to make one more observation about maximal free splittings of F N . Proposition 3.7.
Let T be a maximal free splitting of F N , and let e and e be two edgesin T that do not belong to the same F N -orbit of edges. Assume that equivariantly collapsingeither e or e to a point yields a splitting whose only adjacent F Z maxN -maximal splitting is a i h b i cd h a ih a ih a i h b ih b ih b i Figure 2: An
F Z max -maximal splitting that fails to satisfy the conclusion of Proposition 3.5. T . Then equivariantly collapsing both e and e to points yields a splitting of finite valencein F Z maxN .Proof.
As noticed in the proof of Proposition 3.4, the edges e and e project to loop-edgesin the quotient graph T /F N . The claim follows by noticing that two such edges cannot beadjacent in T /F N , as all vertices of a maximal free splitting have valence . Hence the onlyway of attaching back edges to the splitting obtained by equivariantly collapsing both e and e to points is to attach e and/or e .In contrast to free splittings, an F Z maxN -maximal splitting which contains a nontrivialedge stabilizer fails to satisfy either Proposition 3.4 or Proposition 3.7.
Proposition 3.8.
Let T ∈ F Z maxN be an
F Z maxN -maximal splitting that has some nontrivialedge stabilizer. At least one of the following two points holds:1. There exists an edge e in T such that equivariantly collapsing e to a point yields asplitting of infinite valence in F Z maxN .2. There exist two edges e and e in T such that:(a) Equivariantly collapsing either e or e to a point yields a splitting whose onlyadjacent F Z maxN -maximal splitting is T .(b) Equivariantly collapsing both e and e yields a splitting of infinite valence in F Z maxN . To prove this, we first take a detour to look at cylinders in such splittings. A cylinder in T is a maximal subtree Y in T with the property that all edges of Y have the same stabilizer.Note that an unfoldable edge (as defined in Section 1) is extremal in its associated cylinder(it is adjacent to a leaf vertex of Y ). 10 emma 3.9. Let T ∈ F Z maxN be a splitting that has some nontrivial edge stabilizer. Thereexists a cylinder Y in T with nontrivial edge stabilizers that either contains an unfoldableedge which does not belong to any stabilized tripod, or contains two unfoldable edges indistinct F N -orbits attached to the same vertex.Proof. Suppose otherwise. By Lemma 1.1, the tree T contains an unfoldable edge e . Let Y be the cylinder in T containing e . The edge e belongs to some stabilized tripod in Y , and noedge in Y adjacent to e is unfoldable. Let T (cid:48) be the tree obtained from T by fully unfolding e , and f : T (cid:48) → T the associated fold map. Let Y (cid:48) be the cylinder of T (cid:48) corresponding tothe preimage of Y \ e under f . Outside of the F N –orbit of Y (cid:48) , cylinders of T (cid:48) are mappedisomorphically under f to cylinders of T . We claim that if an edge e (cid:48) with a nontrivialstabilizer in T (cid:48) is unfoldable then f ( e (cid:48) ) is unfoldable in T . This claim, along with the factthat a cylinder of T (cid:48) is either isomorphic to Y (cid:48) or has the same structure as a cylinder of T ,implies that T (cid:48) also does not satisfy the lemma (recall that no edge e (cid:48) adjacent to e in Y isunfoldable). We then obtain a contradiction by using induction on the number of F N -orbitsof edges with nontrivial stabilizers, as a splitting with only one such orbit certainly doessatisfy the lemma. Our claim follows by looking at the peripheral subgroups of the vertexgroups of T (cid:48) (i.e., the set of G v -conjugacy classes of edge groups in a vertex group G v ),as the peripheral subgroups of G v (counted with multiplicity) determine which edges areunfoldable at v .Suppose that e (cid:48) ⊆ T (cid:48) is unfoldable at a vertex v ∈ T (cid:48) . Let v be the vertex of e inthe interior of Y , and v the vertex of e which is a leaf of Y . If f ( v ) is not in the orbitof either v or v then G v = G f ( v ) and these groups have the same peripheral subgroups,with same multiplicity, hence f ( e (cid:48) ) is unfoldable in T . If f ( v ) = v , then as v belongs toa tripod in Y , the group G e is still a peripheral subgroup of G v = G v with multiplicity atleast and the same assertion holds. The remaining case to consider is when the terminalvertex of e (cid:48) ⊆ T (cid:48) satisfies f ( v ) = v . As we have unfolded the edge e at v there is a freefactor decomposition G v = G e ∗ G v , and as e (cid:48) is unfoldable in T (cid:48) there is a free factordecomposition G v = G e (cid:48) ∗ A , with A containing a representative of [ G e (cid:48)(cid:48) ] for each edge e (cid:48)(cid:48) adjacent to v with [ e (cid:48)(cid:48) ] (cid:54) = [ e (cid:48) ] . The free factor decomposition G v = G e ∗ G e (cid:48) ∗ A thenwitnesses the fact that f ( e (cid:48) ) is unfoldable in T . Proof of Proposition 3.8.
By Lemma 3.9 there exists an unfoldable leaf-edge e of a cylinderin T such that either e does not belong to any stabilized tripod, or e is adjacent to anotherunfoldable leaf-edge in the same cylinder. As T is F Z maxN -maximal, Lemma 1.2 tells usthat the edge e has an adjacent vertex v of valence (in the quotient graph T /F N ) suchthat its second adjacent edge e (cid:48) has a trivial stabilizer. Let v (cid:48) be the other vertex of e .If e does not belong to any stabilized tripod, then the argument in the proof of Propo-sition 3.5 applies to show that the first conclusion of the proposition holds.If e belongs to a stabilized tripod, then e is adjacent to another unfoldable edge e in thecorresponding cylinder Y . If the rank of the stabilizer of the common vertex v (cid:48) of e and e is at least , then equivariantly collapsing one of these edges to a point yields a splitting of11nfinite valence, as in the second paragraph of the proof of Proposition 3.5. Otherwise, weclaim that the pair { e, e } satisfies the second conclusion of the proposition. Indeed, as e is unfoldable and T is F Z maxN -maximal, the edge e is also adjacent to an edge with trivialstabilizer. Equivariantly collapsing both e and e to points yields a splitting of infinitevalence by the same arguments as in the last paragraph of the proof of Proposition 3.5. Let T (cid:48) be a splitting obtained by collapsing only one of these orbits of edges. Maximality of T implies that all possible refinements of T (cid:48) are obtained by attaching an edge at v (cid:48) , and as v (cid:48) has cyclic stabilizer, the only way to do this yields the splitting T back. Corollary 3.10.
Any isometry of
F Z maxN preserves
F S N (setwise).Proof. We proceed in the same way as for
F Z N and V S N . Proposition 2.4 tells us that anyisometry preserves the set of maximal splittings. Furthermore, Remark 2.6 tells us that thenumber of edges in a splitting and thus the property of adjacent vertices being collapsesor refinements are also invariant under an isometry. Proposition 3.8 tells us that everynon-free F Z maxN -maximal splitting T either has a one-edge collapse with infinite valence in F Z maxN , or a two-edge collapse with infinite valence such that both intermediate one-edgecollapses have T as their unique maximal refinement. Neither of these properties hold fora maximal free splitting (Propositions 3.4 and 3.7). It follows that any isometry of F Z maxN preserves the set of maximal free splittings, and as free splittings are exactly the splittingsof distance at most one from a maximal free splitting, such an isometry preserves
F S N also. Let
F S (cid:48) N be the graph whose vertices are one-edge free splittings of F N , two vertices beingjoined by an edge if they admit a common refinement. If one adds higher-dimensionalsimplices to F S (cid:48) N in a natural way then F S N becomes the 1-skeleton of the barycentricsubdivision of F S (cid:48) N . The automorphism group of F S (cid:48) N was computed by Aramayona andSouto in [2]. Theorem 4.1. (Aramayona-Souto [2]) For all N ≥ , the natural map from Out ( F N ) tothe isometry group of F S (cid:48) N is an isomorphism. This implies a similar statement for
F S N . Proposition 4.2.
For all N ≥ , the natural map from Out ( F N ) to the isometry group of F S N is an isomorphism.Proof. Let f be an isometry of F S N , and denote by X the subset of F S N consisting of one-edge splittings. Proposition 2.4 implies that f ( X ) = X . In addition, two distinct one-edgesplittings T and T admit a common refinement if and only if d F S N ( T , T ) = 2 . Hence f maps pairs of compatible one-edge splittings to pairs of compatible one-edge splittings, so it12nduces an isometry of F S (cid:48) N . Using Theorem 4.1, we can thus assume, up to precomposing f by an element of Out ( F N ) , that f fixes X pointwise. As every free splitting is characterizedby the set of one-edge splittings that are adjacent to it in F S N (Theorem 1.3), this impliesthat f is the identity map. The claim follows.As we now know that F S N is preserved by isometries of F Z N , F Z maxN , and
V S N ,composing with an appropriate element of Out ( F N ) allows us to restrict our attention toisometries which fix F S N pointwise. F S N pointwise is the identity. Throughout the section, we will denote by G one of the graphs F Z N , F Z maxN or V S N . Wedefine a bad splitting of F N to be a splitting of the form ( F N − ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) , where t ∈ F N is a stable letter of the HNN extension, and w ∈ F N − is not contained in any proper freefactor of F N − . A good splitting is a splitting which is not bad. The following propositiongives a characterization of bad splittings among one-edge splittings. Given w ∈ F N , wedenote by Fill ( w ) the smallest free factor of F N that contains w . Proposition 5.1.
Let T be a one-edge splitting of F N . • If T is good, then there are infinitely many one-edge free splittings that are compatiblewith T . • If T is bad, of the form ( A ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) , then the splitting A ∗ is the only one-edge freesplitting that is compatible with T .Proof. Let T be a good one-edge splitting. The situation when T is a free splitting or aseparating one-edge Z -splitting is covered in the proof of Lemma 2.2, so we assume that T is of the form ( A ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) , where A is a corank one free factor of F N , and w iscontained in some proper free factor of A . Let A (cid:48) be a complementary free factor of Fill ( w ) in A , and let { a (cid:48) , . . . , a (cid:48) i } be a free basis of A (cid:48) . Then for all k ∈ Z , the free splitting ( Fill ( w ) ∗ (cid:104) t (cid:105) ) ∗ (cid:104) a (cid:48) ( w t ) k , . . . , a (cid:48) i (cid:105) is compatible with T . The element a (cid:48) ( w t ) l is elliptic in thissplitting if and only if l = k , so we obtain infinitely many distinct splittings as k varies.Now assume that T is a bad one-edge splitting, of the form ( A ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) , where A is acorank one free factor of F N , and w is not contained in any proper free factor of A . Let S bea free splitting that is compatible with T . Then the common two-edge refinement S (cid:48) of S and T has one of the forms displayed on Figure 3, for some free factors A (cid:48) and B (cid:48) of F N andsome t (cid:48) , t , t ∈ F N . In Cases 1 and 2, slightly unfolding the edge with nontrivial stabilizerin S (cid:48) as in Lemma 1.1 shows that w should be contained in a corank free factor of F N , acontradiction. Hence Case 3 occurs, and as w ∈ A (cid:48) , we necessarily have A (cid:48) = Fill ( w ) . Theclaim follows. 13 ase 1 Case 2 Case 3 h w ih w i h w ih w i h w i A ′ ∗ h w t ′ i A ′ ∗ h w t i A ′ B ′ B ′ A ′ A ′ t t t t t ′ t ′ t ′ Figure 3: The situation in the proof of Proposition 5.1.From Proposition 2.4 we know that the set of one-edge splittings is preserved under anyisometry of G . Proposition 5.1 then implies: Proposition 5.2.
Any isometry of G which restricts to the identity on F S N preserves boththe set of bad one-edge splittings and the set of good one-edge splittings (setwise). We will now prove that any such isometry fixes the set of good one-edge splittingspointwise, by showing that any good one-edge splitting is characterized by the collectionof one-edge free splittings that are compatible with it. The following lemma can be provedby considering common refinements of T and T (cid:48) on a case-by-case basis. Lemma 5.3.
Let T and T (cid:48) be two compatible cyclic splittings of F N . • Any element of F N that fixes an edge of T is elliptic in T (cid:48) . • If T and T (cid:48) are both separating one-edge splittings, then every elliptic subgroup of T (cid:48) is either contained in or contains an elliptic subgroup of T . • If T is a one-edge nonseparating splitting and T (cid:48) is a one-edge separating splitting,then some vertex subgroup of T (cid:48) is contained in a vertex subgroup of T . Proposition 5.4.
Let T and T (cid:48) be two good one-edge Z -splittings of F N . There exists aone-edge free splitting of F N which is compatible with exactly one of the splittings T or T (cid:48) . roof. Up to interchanging T and T (cid:48) , we can assume that either T is separating, or both T and T (cid:48) are nonseparating. Case 1 : The splitting T is separating, of the form A ∗ (cid:104) w (cid:105) ( B ∗ (cid:104) w (cid:105) ) , where A and B are nontrivial proper free factors of F N , and w ∈ A . We denote by { b , . . . , b k } a freebasis of B . Let Fill ( w ) be the smallest free factor of A that contains w , and let A (cid:48) be acomplementary free factor (possibly Fill ( w ) = A and A (cid:48) = { e } ). Let S be a free splitting of F N whose quotient graph is obtained by taking a rose corresponding to a basis of A (cid:48) and arose corresponding to a basis of B and attaching each rose by an edge with trivial stabilizerto a vertex with stabilizer Fill ( w ) . This is displayed in Figure 4. It is compatible with T .If T (cid:48) is not compatible with S , then we are done, so we assume that T (cid:48) is compatible with S . The splitting T (cid:48) is thus obtained from S by first equivariantly inserting an edge e withnontrivial stabilizer (which collapses to the only vertex with nontrivial stabilizer in S ) toget a splitting T (cid:48)(cid:48) , and then collapsing all edges but e in T (cid:48)(cid:48) . This amounts to splittingthe vertex group Fill ( w ) of S over Z , which means inserting in S either a "horizontal"separating edge (Cases 1.1 to 1.4), a loop-edge (Case 1.5) or a "vertical" leaf edge out ofthe graph (Cases 1.6 and 1.7). Thanks to Lemma 1.1, we know that the splitting we gethas one of the following forms. Case 1.1 : The splitting T (cid:48) is of the form ( A (cid:48) ∗ C ) ∗ (cid:104) w (cid:48) (cid:105) [( D ∗ (cid:104) w (cid:48) (cid:105) ) ∗ B ] , where C and D are complementary proper free factors of Fill ( w ) , and w (cid:48) ∈ C .Then T (cid:48) is compatible with ( A (cid:48) ∗ C ) ∗ ( D ∗ B ) . However, if T were compatible with ( A (cid:48) ∗ C ) ∗ ( D ∗ B ) , then the first point of Lemma 5.3 implies that w should be conju-gated into either A (cid:48) ∗ C or D ∗ B , which is not the case. Case 1.2 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ ( C ∗ (cid:104) w (cid:48) (cid:105) )] ∗ (cid:104) w (cid:48) (cid:105) ( D ∗ B ) , where C and D are complementary proper free factors of Fill ( w ) , and w (cid:48) ∈ D .Then the same splitting as in Case 1.1 works. Case 1.3 : The splitting T (cid:48) is of the form ( A (cid:48) ∗ Fill ( w )) ∗ (cid:104) w (cid:48) (cid:105) ( B ∗ (cid:104) w (cid:48) (cid:105) ) = A ∗ (cid:104) w (cid:48) (cid:105) ( B ∗ (cid:104) w (cid:48) (cid:105) ) ,where w (cid:48) ∈ Fill ( w ) and w (cid:48) (cid:54) = w ± .Up to exchanging the roles of T and T (cid:48) , we can assume that w / ∈ (cid:104) w (cid:48) (cid:105) . Then T is compatiblewith A ∗ (cid:104) b w, b , . . . , b k (cid:105) . However, if T (cid:48) were compatible with A ∗ (cid:104) b w, b , . . . , b k (cid:105) , thenby the second point of Lemma 5.3, the subgroup (cid:104) b w, b , . . . , b k (cid:105) should either contain orbe contained in B ∗ (cid:104) w (cid:48) (cid:105) (these vertex stabilizers are malnormal and have nontrivial inter-section). This is not the case, as w (cid:48) (cid:54)∈ (cid:104) b w, b , . . . , b k (cid:105) and b w (cid:54)∈ B ∗ (cid:104) w (cid:48) (cid:105) . Case 1.4 : The splitting T (cid:48) is of the form ( A (cid:48) ∗ (cid:104) w (cid:48) (cid:105) ) ∗ (cid:104) w (cid:48) (cid:105) ( Fill ( w ) ∗ B ) , with w (cid:48) ∈ Fill ( w ) .If Fill ( w ) = (cid:104) w (cid:105) , then T (cid:48) is of the form ( B ∗ (cid:104) w (cid:105) ) ∗ (cid:104) w (cid:48) (cid:105) ( A (cid:48) ∗ (cid:104) w (cid:48) (cid:105) ) (while T is of the form ( B ∗ (cid:104) w (cid:105) ) ∗ (cid:104) w (cid:105) ( A (cid:48) ∗ (cid:104) w (cid:105) ) ), and the claim follows from the argument of Case 1.3. Otherwise,let w (cid:48)(cid:48) ∈ Fill ( w ) (cid:114) (cid:104) w (cid:105) . Then T (cid:48) is compatible with A ∗ (cid:104) b w (cid:48)(cid:48) , b , . . . , b k (cid:105) (they have a15ommon two-edge refinement of the form ( A (cid:48) ∗ (cid:104) w (cid:48) (cid:105) ) ∗ (cid:104) w (cid:48) (cid:105) Fill ( w ) ∗ (cid:104) b w (cid:48)(cid:48) , b , . . . , b k (cid:105) ), while T is not by the same argument as in Case 1.3. Case 1.5 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ ( C ∗ (cid:104) w (cid:48) (cid:105) t ) ∗ B ] ∗ (cid:104) w (cid:48) (cid:105) , where C is a corankone free factor of Fill ( w ) , and w (cid:48) ∈ C , and t ∈ Fill ( w ) .Then the splitting ( A (cid:48) ∗ B ∗ C ) ∗ is compatible with T (cid:48) , but not with T since w is notconjugated into A (cid:48) ∗ B ∗ C . Case 1.6 : The splitting T (cid:48) is of the form ( A (cid:48) ∗ B ∗ C ) ∗ (cid:104) w (cid:48) (cid:105) ( C (cid:48) ∗(cid:104) w (cid:48) (cid:105) ) , where C ∗ C (cid:48) = Fill ( w ) ,and w (cid:48) ∈ C .In particular, we have C (cid:48) (cid:54) = { e } , and the splitting ( A (cid:48) ∗ B ∗ C ) ∗ C (cid:48) is compatible with T (cid:48) ,but not with T since w is neither conjugated into A (cid:48) ∗ B ∗ C , nor into C (cid:48) . Case 1.7 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ B ∗ ( C ∗ (cid:104) w (cid:48) (cid:105) )] ∗ (cid:104) w (cid:48) (cid:105) C (cid:48) , where C ∗ C (cid:48) = Fill ( w ) ,and w (cid:48) ∈ C (cid:48) .If A (cid:48) = { e } and C (cid:48) = Fill ( w ) , then T (cid:48) is of the form Fill ( w ) ∗ (cid:104) w (cid:48) (cid:105) ( B ∗ (cid:104) w (cid:48) (cid:105) ) (while T is ofthe form Fill ( w ) ∗ (cid:104) w (cid:105) ( B ∗ (cid:104) w (cid:105) ) ), so the claim follows from the argument of Case 1.3.Otherwise, the same splitting as in Case 1.6 works by the second point of Lemma 5.3.Indeed, in this case, we have B (cid:54) = { e } and C (cid:54) = Fill ( w ) . The group A = A (cid:48) ∗ Fill ( w ) isneither contained in nor contains A (cid:48) ∗ B ∗ C , so the conclusion follows. Case 2 : Both splittings T and T (cid:48) are nonseparating. By Lemma 1.1, the splitting T is of the form ( A ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) for some corank one free factor A of F N and some w ∈ A .We denote by Fill ( w ) the smallest free factor of A that contains w , and we let A (cid:48) be acomplementary free factor in A . As T is assumed to be good, we have Fill ( w ) (cid:54) = A , andhence A (cid:48) (cid:54) = { e } . We denote by { a (cid:48) , . . . , a (cid:48) k } a free basis of A (cid:48) . Let S be the free splittingdisplayed on Figure 5, it is compatible with T . Again, we may assume that T (cid:48) is compatiblewith S (otherwise we are done). Passing from S to T (cid:48) again requires inserting an edge with Z stabilizer to get a new tree T (cid:48)(cid:48) , then equivariantly collapsing its complement in T (cid:48)(cid:48) . As T (cid:48) is nonseparating, the inserted edge can either be a separating edge lying on the looplabelled by t in S (which leads to Cases 2.1 to 2.4) or a loop-edge (Case 2.5). Case 2.1 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ C ∗ ( D ∗ (cid:104) w (cid:48) (cid:105) ) t ] ∗ (cid:104) w (cid:48) (cid:105) , where C and D are two complementary free factors of Fill ( w ) , and w (cid:48) ∈ C .We have w (cid:48) (cid:54) = w ± (otherwise C = Fill ( w ) and T (cid:48) = T ). Up to exchanging the roles of T and T (cid:48) , we can assume that w / ∈ (cid:104) w (cid:48) (cid:105) . In this case, the splitting T (cid:48)(cid:48) := ( Fill ( w ) ∗ (cid:104) t (cid:105) ) ∗(cid:104) a (cid:48) w t , a (cid:48) , . . . , a (cid:48) k (cid:105) is compatible with T . However, if T (cid:48)(cid:48) were also compatible with T (cid:48) , bythe third point of Lemma 5.3 one of the vertex stabilizers of T (cid:48)(cid:48) should be elliptic in T (cid:48) .This is not the case, since both t and a (cid:48) w t are hyperbolic in T (cid:48) . Case 2.2 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ C ∗ ( D ∗ (cid:104) w (cid:48) (cid:105) ) t − ] ∗ (cid:104) w (cid:48) (cid:105) , where C and D ill( w ) A ′ B Figure 4: The splitting S in Case 1 of the proof of Proposition 5.4.are two complementary free factors of Fill ( w ) , and w (cid:48) ∈ C .Then the same splitting as in Case 2.1 works. Case 2.3 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ ( C ∗ (cid:104) w (cid:48) (cid:105) ) ∗ D t ] ∗ (cid:104) w (cid:48) (cid:105) , where C and D are two complementary free factors of Fill ( w ) , and w (cid:48) ∈ D .We may also assume C ∗ (cid:104) w (cid:48) (cid:105) (cid:54) = Fill ( w ) , otherwise we are in a particular case of Case 2.1.Let w (cid:48)(cid:48) ∈ Fill ( w ) (cid:114) ( C ∗ (cid:104) w (cid:48) (cid:105) ) . Then the splitting T (cid:48)(cid:48) := ( Fill ( w ) ∗ (cid:104) t (cid:105) ) ∗ (cid:104) a (cid:48) w (cid:48)(cid:48) , a (cid:48) , . . . , a (cid:48) k (cid:105) iscompatible with T . If it were also compatible with T (cid:48) , the third point of Lemma 5.3 wouldimply that one of the edge groups of T (cid:48)(cid:48) is elliptic in T (cid:48) . This is not the case, since both t and a (cid:48) w (cid:48)(cid:48) are seen to be hyperbolic in T (cid:48) . Case 2.4 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ ( C ∗ (cid:104) w (cid:48) (cid:105) ) ∗ D t − ] ∗ (cid:104) w (cid:48) (cid:105) , where C and D are two complementary free factors of Fill ( w ) , and w (cid:48) ∈ D .The same splitting as in Case 2.3 works. Case 2.5 : The splitting T (cid:48) is of the form [ A (cid:48) ∗ ( C ∗ (cid:104) t (cid:105) ) ∗ (cid:104) w (cid:48) t (cid:48) (cid:105) ] ∗ (cid:104) w (cid:48) (cid:105) , where C is a corankone free factor of Fill ( w ) , and w (cid:48) ∈ C , and t (cid:48) ∈ Fill ( w ) .Let w (cid:48)(cid:48) ∈ Fill ( w ) (cid:114) ( C ∗ (cid:104) w (cid:48) t (cid:48) (cid:105) ) . Again, the splitting ( Fill ( w ) ∗ (cid:104) t (cid:105) ) ∗ (cid:104) a (cid:48) w (cid:48)(cid:48) , . . . , a (cid:48) k (cid:105) iscompatible with T , but not with T (cid:48) since both w (cid:48)(cid:48) and a (cid:48) w (cid:48)(cid:48) are hyperbolic in T (cid:48) . Proposition 5.5.
Any isometry of G that restricts to the identity on F S N fixes the set ofgood one-edge splittings pointwise.Proof. By Proposition 5.2, any isometry f of G that restricts to the identity on F S N preserves the set X of good one-edge splittings. If T and T (cid:48) are two distinct good one-edge17 ill( w ) A ′ t Figure 5: The splitting S in Case 2 of the proof of Proposition 5.4.splittings, Proposition 5.4 implies that the set of one-edge free splittings at distance at most from T and the set of one-edge free splittings at distance at most from T (cid:48) are distinct.Hence f fixes X pointwise. Proposition 5.6.
Let T and T (cid:48) be two distinct bad one-edge cyclic splittings of F N . Thenthere exists a good one-edge cyclic splitting T (cid:48)(cid:48) of F N which is compatible with exactly oneof the splittings T and T (cid:48) . If in addition T and T (cid:48) are maximally-cyclic, then T (cid:48)(cid:48) can bechosen to be maximally-cyclic.Proof. Let T and T (cid:48) be two bad one-edge splittings. Proposition 5.1 implies that there isexactly one free splitting S (resp. S (cid:48) ) which is compatible with T (resp. T (cid:48) ). If S (cid:54) = S (cid:48) ,then we are done, so we assume that S = S (cid:48) . The splitting T is of the form ( A ∗ (cid:104) w t (cid:105) ) ∗ (cid:104) w (cid:105) for some corank one free factor A of F N and some w ∈ A . Case 1 : The splitting T (cid:48) is of the form ( A ∗ (cid:104) w (cid:48) t (cid:105) ) ∗ (cid:104) w (cid:48) (cid:105) .Up to exchanging the roles of T and T (cid:48) , we can assume that w (cid:48) / ∈ (cid:104) w (cid:105) . Denoting by T (cid:48)(cid:48) the separating (and hence good) splitting A ∗ (cid:104) w (cid:105) (cid:104) w, t (cid:105) , we get that T (cid:48)(cid:48) is compatible with T , but not with T (cid:48) . Indeed, if it were, then in a common refinement of T (cid:48) and T (cid:48)(cid:48) , the axisof t should meet the fixed point set of w (cid:48) , which leads to a contradiction because w (cid:48) and t belong to distinct elliptic subgroups of T (cid:48)(cid:48) . Case 2 : The splitting T (cid:48) is of the form ( A ∗ (cid:104) w (cid:48) t − (cid:105) ) ∗ (cid:104) w (cid:48) (cid:105) .Let w (cid:48)(cid:48) ∈ A be contained in some proper free factor of A (this exists because N ≥ ). Thesplitting ( A ∗ (cid:104) w (cid:48)(cid:48) t − (cid:105) ) ∗ (cid:104) w (cid:48)(cid:48) (cid:105) is then a good splitting which is compatible with T , but notwith T (cid:48) . (This can be seen by looking at the possible forms of a two-edge refinement of T (cid:48) with this splitting, and reaching a contradiction in each case.)18 roposition 5.7. Any isometry of G that restricts to the identity on F S N fixes the set ofbad one-edge splittings of F N pointwise.Proof. Let f be an isometry of G that restricts to the identity on F S N . It follows fromProposition 5.2 that f preserves the set of bad one-edge splittings of F N . If T and T (cid:48) aretwo bad one-edge splittings in G , then Proposition 5.6 implies that the set of good one-edgesplittings at distance from S in G is distinct from the set of good one-edge splittings atdistance from T (cid:48) in G . The claim thus follows from Proposition 5.5. Proof of Theorem A.
Let G be one of the graphs F Z N , F Z maxN or V S N , and let f be anisometry of G . Corollary 3.3 (for the case where G = F Z N ), Corollary 3.6 (when G = V S N )or Corollary 3.10 (when G = F Z maxN ) show that f preserves F S N setwise. Proposition 4.2then implies that up to composing f with an element of Out ( F N ) , we may assume that f restricts to the identity on F S N , and we want to show that f = id. The set X (resp. X )of good (resp. bad) one-edge splittings in G is fixed pointwise by f (Propositions 5.5 and5.7). Using Theorem 1.3, we get that for all T, T (cid:48) ∈ G (cid:114) ( X ∪ X ) , the set of elements in X ∪ X at distance from T differs from the set of elements at distance from T (cid:48) , so f also fixes G (cid:114) ( X ∪ X ) pointwise. Theorem A follows. Denote by ( F Z N ) (cid:48) (respectively ( F Z maxN ) (cid:48) ) the graph whose vertices are the equivalenceclasses of one-edge cyclic (resp. maximally-cyclic) splittings of F N , in which two verticesare joined by an edge whenever they are compatible. These are dual versions of the graphswe have been dealing with so far. Proposition 6.1.
For all N ≥ , the natural maps from Out ( F N ) to the isometry groupsof ( F Z N ) (cid:48) and ( F Z maxN ) (cid:48) are isomorphisms.Proof. Let G be one of the graphs F Z N or F Z maxN , and G (cid:48) be its "dual" version. Let f bean isometry of G (cid:48) . Let T ∈ G . The set of one-edge splittings at distance at most from T in G is a collection of pairwise compatible one-edge splittings, hence it is mapped by f to a collection C of pairwise compatible one-edge splittings in G . Thanks to Theorem 1.3,we know that the splittings in C have a common refinement in G , which is the unique tree T (cid:48) ∈ G such that the set of one-edge splittings at distance at most from T (cid:48) is equal to C .Hence f induces a map f (cid:48) : G → G , mapping T to T (cid:48) . This is an isometry: it is a bijectionbecause f is, and any collapse (resp. refinement) of a splitting is mapped by f to a collapse(resp. refinement) of its f -image. By Theorem A, there exists Φ ∈ Out ( F N ) such that forall T ∈ G , we have f (cid:48) ( T ) = T · Φ . This holds in particular for one-edge splittings of F N , forwhich f ( T ) = f (cid:48) ( T ) . Hence f is induced by an element of Out ( F N ) . Remark . One might turn the graph ( F Z maxN ) (cid:48) into a higher-dimensional complex (cid:92) F Z maxN (or (cid:91)
V S N ) by adding a k -simplex for each maximally-cyclic (or very small) splitting of F N k orbits of edges. Thanks to Theorem 1.4, we know that the complexes (cid:92) F Z maxN and (cid:91)
V S N are finite-dimensional (whereas the same construction would lead to aninfinite-dimensional complex in the case of general cyclic splittings of F N ). Any simplicialautomorphism of either (cid:92) F Z maxN or (cid:91) V S N restricts to an isometry of its one-skeleton, hencethe groups of simplicial automorphisms of (cid:92) F Z maxN and (cid:91)
V S N also coincide with Out ( F N ) .The complex (cid:92) F Z maxN is flag by Theorem 1.3, while (cid:91)
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