On finitely generated normal subgroups of Kähler groups
OOn finitely generated normal subgroups of K¨ahler groups
Francisco Nicol´asFebruary 2021
Abstract
We prove that if a surface group embeds as a normal subgroup in a K¨ahler group and theconjugation action of the K¨ahler group on the surface group preserves the conjugacy class ofa non-trivial element, then the K¨ahler group is virtually given by a direct product, where onefactor is a surface group. Moreover we prove that if a one-ended hyperbolic group with infiniteouter automorphism group embeds as a normal subgroup in a K¨ahler group then it is virtuallya surface group. More generally we give restrictions on normal subgroups of K¨ahler groupswhich are amalgamated products or HNN extensions.
Let G be a finitely generated group. Bass-Serre Theory (see [20] and [21]) establishes a dictionarybetween decompositions of G as an amalgamated product or an HNN extension and actions of G ona simplicial tree without inversions which are transitive on the set of edges. Of course, the theoryalso deals with more complicated graphs of groups but we will mainly deal with amalgamatedproducts and HNN extensions. If G acts on a tree T and we have finitely generated groups Γ and Q that fit in a short exact sequence1 (cid:47) (cid:47) G (cid:47) (cid:47) Γ (cid:47) (cid:47) Q (cid:47) (cid:47) , (1)one can ask whether the action of G on T can be extended to Γ. If the center of G acts trivially on T , we obtain an induced action of the group Inn( G ) of inner automorphisms of G on T . As before,one can ask whether this induced action can be extended to a larger group of automorphismsof G . Of course both questions are related, since for groups Γ and G as in (1) the conjugationaction of Γ on G induces a map Γ → Aut( G ). Several authors have studied the latter question.Karrass, Pietrowski and Solitar [13] studied the case of an amalgamated product A ∗ C B , where C is maximal along all its conjugates in A and B . Pettet [17] studied the general situation ofa graph of groups with a more restrictive condition on the edge stabilizers of its Bass-Serre tree( “edge group incomparability hypothesis” ) which is equivalent to the conjugate maximal condition of Karrass, Pietrowski and Solitar when there is one orbit of edges. A particular case of thissituation was studied by Gilbert, Howie, Metaftsis and Raptis [10] where they proved that theaction of the Baumslag-Solitar group on its Bass-Serre tree can be extended to its whole group ofautomorphisms. In the context of a one-ended torsion-free hyperbolic group G , Sela [18] provedthe existence of a “canonical” tree T on which G acts. By studying the action of Aut( G ) on ∂G ,Bowditch [3] proved that the action of G on T can be extended to Aut( G ).We will apply these results to study finitely generated normal subgroups of K¨ahler groups. Re-call that a group is called a K¨ahler group if it can be realized as the fundamental group of a compact1 a r X i v : . [ m a t h . G T ] F e b ¨ahler manifold, a classical reference on this subject is [1]. Some examples of K¨ahler groups areabelian groups of even rank, surface groups (groups that can be realized as the fundamental groupof a closed hyperbolic surface) and uniform lattices of P U ( n,
1) (the group of holomorphic isome-tries of complex hyperbolic n -space). The main ingredient to apply these results about actionson trees that extend to a group of automorphisms is a classical result of Gromov and Schoen [11]about K¨ahler groups acting on trees.First, we will study short exact sequences as in (1), where Γ is a K¨ahler group and G is a surfacegroup, i.e. , when G can be realized as the fundamental group of a closed hyperbolic surface S .The first main result of this work is the folllowing: Theorem A
Suppose that the conjugation action of Γ on π ( S ) preserves the conjugacy class ofa simple closed curve in S . Then there is a finite index subgroup Γ of Γ containing π ( S ) suchthat the restricted short exact sequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ (cid:47) (cid:47) Q (cid:47) (cid:47) splits as a direct product (where Q is the image of Γ in Q ). A particular case arises when a K¨ahler extension by a surface group has an abelian quotient. In thatcase, using classical results on the mapping class group we have that for such a short exact sequence,the hypothesis of Theorem A is virtually fulfilled if the rank of the image of Q → Out( π ( S )) isgreater than 1. From this we can deduce that a map ψ : Q → Out( π ( S )) with Q any abeliangroup, is the monodromy map of a short exact sequence as in (1), with Γ a K¨ahler group and G = π ( S ), only if it has finite image. This result is due to Bregman and Zhang (see [4]) whoestablished it using different techniques. We recover it as a corollary of Theorem A (see Section4.1).Going back to the case of an arbitrary quotient Q , Theorem A says that if the image of themonodromy map Q → Out( π ( S )) is infinite, it cannot preserve any admissible set of S ( S )(see section 4.1 for the definition of admissible). It is well known that this conclusion holdswhen the short exact sequence is induced by a surjective holomorphic submersion (which is nota locally trivial holomorphic fiber bundle) between compact complex manifolds, whose fibers arehomeomorphic to the closed hyperbolic surface S and whose base is either projective or a Riemannsurface of finite type. This is a theorem due to Shiga [22]. Hence, our theorem can be seen as ageneralization of Shiga’s result (in the case of compact bases).We now go back to the study of extensions as in (1) where G need not be a surface group. Thefolllowing three results are an application of our study on K¨ahler groups and the works [3, 10, 13,17, 18]. Recall that a group H is indecomposable if for any decomposition H = H ∗ H as a freeproduct, H or H is trivial. Theorem B
Let G be a group that splits as a non-trivial free product A ∗ B with A and B in-decomposable, not infinite cyclic and non isomorphic to each other. Then G does not embed as anormal subgroup in a K¨ahler group. Theorem C
Let G be the Baumslag-Solitar group (cid:104) x, t | tx p t − = x q (cid:105) , where p, q are integers with p, q > and such that neither one is a multiple of the other. Then G does not embed as a normal subgroup in a K¨ahler group. Finally, our last result deals with the case where the normal subgroup G is a one-ended hyper-bolic group. 2 heorem D Suppose that G is a one-ended hyperbolic group without torsion that embeds as anormal subgroup in a K¨ahler group Γ . If Out( G ) is infinite, then G is virtually a surface group. We observe that in this last theorem there is no hypothesis on the monodromy map Γ → Out( G ).We only need to know that Out( G ) is infinite.The structure of this work is the following. In Section 2, we explain how to extend the actionof a group on a tree to a subgroup of its group of automorphisms. In particular we study the caseof surface groups. In section 3, we apply these results to the study of K¨ahler groups admitting asa normal subgroup a group acting on a tree. Theorems A, B, C and D are proved in section 3.2.Finally, in Section 4 we study the monodromy map of a K¨ahler extension by a surface group andestablish variations on Theorem A. Notations.
Suppose that G acts on a tree T and let e and v be an edge and a vertex of T respectively. We will denote by G e and G v the subgroups of G given by the respective stabilizersof e and v .The dictionary established by Bass-Serre theory is given by the following results (see [20, 21]): Theorem 1 (Serre)
Let G act on a tree T without inversions in such a way that the action istransitive on the set of edges. Let e = ( v , v ) be an edge of T .1. If v and v are in different orbits, then G splits as the amalgamated product G v ∗ G e G v .2. If v and v are in the same orbit, then G splits as the HNN extension G v ∗ G e ,θ where θ : G e → G v is the monomorphism given by the conjugation by an element t in G \ G v thatsends v to v . Theorem 2 (Serre)
Let G be a group that splits as an amalgamated product or as an HNNextension. Then, G acts on a tree T without inversions and this action is transitive on the set ofedges.1. If G splits as the amalgamated product A ∗ C B , then the set of vertices of T is the disjointunion of left cosets G/A (cid:116)
G/B and the set of edges of T is the set of left cosets G/C . Theadjacency is given by the maps
G/C → G/A and
G/C → G/B .2. If G splits as the HNN extension A ∗ C,θ , where θ : C → A is the monomorphism given by theconjugation by an element t in G \ A , then the set of vertices of T is the set of left cosets G/A and the set of edges of T is the set of left cosets G/C . The adjacency is given by the maps ι : G/C → G/A and ι : G/C → G/A that send xC to xA and xC to xt − A respectively. In both cases we will say that G splits over the subgroup C . The tree associated in this way toa group that splits over a subgroup C is called the Bass-Serre tree of G . The action of G on itsBass-Serre tree T has the following properties:1. It is minimal, i.e. there is no G -invariant proper subtree in the Bass-Serre tree T .3. If C is properly contained in A , then the action on the boundary of the tree has no fixedpoints. Recall that the boundary of a tree is given by the set of infinite paths withoutbacktracking starting at a fixed point of the tree (this is one definition of the boundary of atree among many others).3. If G splits as an amalgamated product A ∗ C B such that [ A : C ] > B : C ] >
2, orif G splits as an HNN extension A ∗ C,θ such that [ A : C ] > A : θ ( C )] >
1, then theBass-Serre tree of G is not a line. Let G be a finitely generated group that splits as an amalgamated product or an HNN extensionand let T be its Bass-Serre tree. If the center Z ( G ) of G acts trivially on T , the action of G on T factors through the quotient G/Z ( G ). Since this quotient is isomorphic to the group Inn( G ) ofinner automorphisms of G , we obtain an induced action of Inn( G ) on T . The aim of this sectionis to extend this induced action of Inn( G ) on T to a larger group of automorphisms of G . Definition 3
We define the subgroup
Aut T ( G ) of Aut( G ) by the following property: an automor-phism ϕ : G → G is an element of Aut T ( G ) if for any edge e = ( v , v ) of T there is an element x in G such that ϕ ( G v ) = xG v x − , ϕ ( G v ) = xG v x − and ϕ ( G e ) = xG e x − . Lemma 4
Suppose that the stabilizer of any vertex of T is its own normalizer and let ϕ : G → G be an element of Aut T ( G ) . Then, ϕ induces an isometry ϕ : T → T such that for all g in Gϕ ( g · ) = ϕ ( g ) · ϕ ( ) . (2) Moreover, if ϕ is the inner automorphism given by the conjugation by x , then ϕ is given by theaction of x on T . Remark 5
The hypothesis of Lemma 4 on the vertex stabilizers implies that the center of G actstrivially on T .Proof of Lemma 4. Let us fix an edge e = ( v , v ). We warn the reader that the followingconstruction of ϕ depends on the choice of e . We write ϕ ( G v ) = xG v x − , ϕ ( G v ) = xG v x − and ϕ ( G e ) = xG e x − . We define an isometry ϕ : T → T as follows. If v is a vertex of T we write v = g · v i for some index i and define ϕ ( v ) = ϕ ( g ) x · v i . If f is an edge of T we write f = g · e and define ϕ ( f ) = ϕ ( g ) x · e. These definitions are independent of the choice of the element g . In the case of an HNN extension,we must check that the action on a vertex v is independent on whether we represent v as the imageof v or v (as v and v are in the same orbit). But if v = t · v one checks that t − x − ϕ ( t ) x normalizes G v , and thus by the hypothesis of the lemma we get that t − x − ϕ ( t ) x lies in G v .Using this observation one proves that ϕ ( v ) is well defined. In this way we obtain two bijections,4ne of the set of vertices of T and one of the set of edges of T , which define an isometry of T . Ifthere is another element y in G such that ϕ ( G v ) = yG v y − , ϕ ( G v ) = yG v y − and ϕ ( G e ) = yG e y − then x − y is in the normalizer of G v and G v . Since G v and G v are their own normalizers weget that x − y is in the intersection of G v and G v , which is G e . This implies that ϕ is well definedand it has the desired property ϕ ( g · ) = ϕ ( g ) · ϕ ( ) . Finally if ϕ is the inner automorphism given by the conjugation by x we get that ϕ ( g · e ) = ϕ ( g ) x · e = ( xgx − ) x · e = x · ( g · e ) . Similarly, for i = 1 , ϕ ( g · v i ) = x · ( g · v i ), which implies that ϕ is given by theaction of x on T . (cid:50) Definition 6
An isometry ϕ : T → T induced by an automorphism ϕ of G that satisfies (2) iscalled G -compatible. An extension of the induced action of Inn( G ) on T to a subgroup Λ of Aut( G ) is called G -compatible if every automorphism of Λ defines a G -compatible isometry of T . Lemma 7
Suppose that the stabilizer of any vertex of T is its own normalizer. Then the inducedaction of Inn( G ) on T extends to a G -compatible action of Aut T ( G ) on T .Proof. Let us fix an edge e = ( v , v ). By Lemma 4, it suffices to show that for all ϕ, ψ in Aut T ( G )the isometries ϕ and ψ defined as before satisfy: ϕ − = ϕ − and ϕ ◦ ψ = ϕ ◦ ψ. Let ϕ be an element in Aut T ( G ) such that ϕ ( G v ) = xG v x − , ϕ ( G v ) = xG v x − and ϕ ( G e ) = xG e x − . Then, if y = ϕ − ( x − ) we get that ϕ − ( G v ) = yG v y − , ϕ − ( G v ) = yG v y − and ϕ − ( G e ) = yG e y − . Therefore the isometry ϕ − is given by ϕ − ( g · v i ) = ϕ − ( g ) y · v i for i = 1 , ϕ − ( g · e ) = ϕ − ( g ) y · e. A direct computation shows that ϕ − = ϕ − .Now, let ϕ and ψ be two elements of Aut( G ) such that ϕ ( G v ) = xG v x − , ϕ ( G v ) = xG v x − and ϕ ( G e ) = xG e x − , and ψ ( G v ) = yG v y − , ψ ( G v ) = yG v y − and ψ ( G e ) = yG e y − . Then, if z = ψ ( x ) y we get that ψ ◦ ϕ ( G v ) = zG v z − , ψ ◦ ϕ ( G v ) = zG v z − and ψ ◦ ϕ ( G e ) = zG e z − . ψ ◦ ϕ is given by ψ ◦ ϕ ( g · v i ) = ψ ◦ ϕ ( g ) z · v i for i = 1 , ψ ◦ ϕ ( g · e ) = ψ ◦ ϕ ( g ) z · e. Once again, a direct computation shows that ψ ◦ ϕ = ψ ◦ ϕ . (cid:50) Note that the isometry ϕ : T → T defined in Lemma 4 depends on the selected edge e = ( v , v )when ϕ is not an inner automorphism of G . Therefore, the action of Aut T ( G ) on T depends aswell on this edge. Definition 8
We denote by Λ T be the subgroup of Aut( G ) that preserves the conjugacy class ofeach vertex stabilizer and each edge stabilizer of the Bass-Serre tree T of G , i.e. an automorphism ϕ of G is an element of Λ T if for any edge e = ( v , v ) there are elements x, y, z in G such that ϕ ( G v ) = xG v x − , ϕ ( G v ) = yG v y − and ϕ ( G e ) = zG e z − . It is clear that Aut T ( G ) is always contained in Λ T . In the following, we will see some situationswhere both groups almost coincide. Let γ be a simple closed curve in a closed hyperbolic surface S . If we denote by C the cyclicsubgroup of π ( S ) generated by the homotopy class of γ , we have that π ( S ) splits over the cyclicsubgroup C as follows. Let us fix a base point x in γ . If γ is a separating curve, we will denote by A and B the fundamental groups of the surfaces obtained by cutting S along γ . Similarly, if γ isnon-separating, we will denote by A the fundamental group of the surface obtained by cutting S along γ . All fundamental groups are based at x . In the case where γ is not separating, we mustchoose a “copy” of x in the surface obtained by cutting S along γ .If γ is a separating curve, π ( S ) splits as the amalgamated product A ∗ C B . If γ is a non-separating curve, π ( S ) splits as the HNN extension A ∗ C,θ , where θ : C → A is the monomorphismgiven by the conjugation by an element t in π ( S ) \ A . More precisely, t is the homotopy class ofa simple closed curve in S that becomes a path in S \ γ joining the boundary components. Remark 9
Given a simple closed curve γ in a closed hyperbolic surface S as before, there existsa “dual tree” associated to γ that embeds in H . This tree coincides with the Bass-Serre treeassociated to the splitting of π ( S ) over C and it can be constructed as follows. Let us assumethat γ is a closed geodesic in S (every non-nullhomotopic simple closed curve in S is homotopicto a unique closed geodesic) and let p : H → S be the universal covering space of S . We denoteby L the set of bi-infinite geodesics in H , whose images under p are equal to γ . We say that twoconnected components of H \ L are related if they are separated by an element of L . This defines asymmetric binary relation on H \ L . The set of vertices of the dual tree is given by the connectedcomponents of H \ L and the adjacency is given by this symmetric binary relation, i.e., the set ofedges of the dual tree is given by L . The action of π ( S ) on H induces an action of π ( S ) on thedual tree, which is consistent with the action of π ( S ) on T . Lemma 10
Let γ be a simple closed curve in S and let T be the Bass-Serre tree of π ( S ) associatedto γ . The group of automorphisms of π ( S ) that preserves the conjugacy class of γ contains Aut T ( π ( S )) as a subgroup of index at most . Hence Λ T contains Aut T ( π ( S )) as a subgroup ofindex at most . roof. Let Γ be the subgroup of Aut( π ( S )) that preserves the conjugacy class of γ . We have theinclusions: Aut T ( π ( S )) ⊂ Λ T ⊂ Γ . Hence the second point of the lemma follows from the first one. By the Dehn-Nielsen-Baer Theorem(see [7]) we have an isomorphism between the (extended) mapping class group of S and the outerautomorphism group of π ( S ) M CG ( S ) = Dif f ± ( S ) /Dif f ( S ) → Out( π ( S )) = Aut( π ( S )) / Inn( π ( S )) , which takes the class of a diffeomorphism f : S → S to the automorphism induced by f on π ( S )(well-defined up to conjugacy). We will prove that any automorphism in Γ which is induced by anorientation preserving diffeomorphism of S lies in Aut T ( π ( S )). Let us fix an automorphism ϕ of π ( S ) contained in Γ and an orientation preserving diffeomorphism f : S → S inducing ϕ .Now, let { f t : S → S } ≤ t ≤ be a path of diffeomorphisms of S such that f is the identity on S and f ◦ γ = f ◦ γ (we think of γ as a parametrized curve). Therefore, we get that f − ◦ f isan orientation preserving diffeomorphism of S fixing γ pointwise. Hence, f − ◦ f preserves theconnected components of S \ γ and the induced map on fundamental groups( f − ◦ f ) ∗ : π ( S ) → π ( S )preserves the cyclic subgroup C . Therefore ( f − ◦ f ) ∗ preserves as well the fundamental group ofthe connected components of S \ γ and hence it lies in Aut T ( π ( S )). Finally, since f and f − ◦ f define the same class in M CG ( S ), we get that ϕ and ( f − ◦ f ) ∗ define the same class in Out( π ( S ))which implies the result. (cid:50) Recall that a subgroup H of a group G is called malnormal if xHx − ∩ H is trivial for all x in G \ H . The fundamental groups of the surfaces obtained by cutting S along γ are malnormal in π ( S ). This implies that the stabilizer of any vertex of T is its own normalizer. Finally, since thecenter of a surface group is trivial, we obtain the following result as a consequence of Lemmas 7and 10. Proposition 11
The action of π ( S ) on T extends to a subgroup of index at most of the groupof automorphisms of π ( S ) that preserves the conjugacy class of γ . Moreover, this extension is π ( S ) -compatible. Definition 12
Let F be a family of subgroups of G . We say that F is maximal if for any subgroups H, K in F , K < H implies K = H . Let G be a group acting on a tree T and let v be a vertex of T . We say that an edge of T is v -incident if one of its vertices is v . We will be interested in the following conditions: C1 The stabilizer of an edge of T is properly contained in the stabilizers of its vertices. C2 For any vertex v in T the family of stabilizers of v -incident edges is maximal. Remark 13
Condition C1 implies that the stabilizer of any vertex is its own normalizer andtherefore the center of G acts trivially on T (see Remark 5). Theorem 14 (Karrass, Pietrowski, Solitar, [13])
Let G be a group that splits as an amalga-mated product A ∗ C B and let T be its Bass-Serre tree. If the action of G on T satisfies conditions C1 and C2 , then the induced action of Inn( G ) on T extends to a G -compatible action of Λ T on T . heorem 15 (Pettet, [17]) Let G be a group that splits as an HNN extension A ∗ C,θ and let T be its Bass-Serre tree. If the action of G on T satisfies conditions C1 and C2 , then the inducedaction of Inn( G ) on T extends to a G -compatible action of Λ T on T . Remark 16
The original statements of Theorems 14 and 15 only give information about thedecomposition of Λ T as an amalgamated product in the case of Theorem 14, and as an HNNextension in the case of Theorem 15, but we can deduce from the proofs of these results than suchdecompositions are induced by an action of Λ T on T that extends the one of G on T . In [17], Pettet gives a more general result than the one stated above. Pettet studied groupsthat split as the fundamental group of a graph of groups with a more restrictive condition on theedge stabilizers of its Bass-Serre tree ( edge group incomparability hypothesis ), which coincides withthe maximality of the families of v -incident edge stabilizers when G splits as an HNN extension.Gilbert, Howie, Metaftsis and Raptis studied the particular situation when the edge stabilizers arecyclic. In particular they proved the following result on Baumslag-Solitar groups: Theorem 17 (Gilbert, Howie, Metaftsis, Raptis)
Let G be the Baumslag-Solitar group (cid:104) x, t | tx p t − = x q (cid:105) , where p, q are integers with p, q > and such that neither one is a multiple of the other. Then theinduced action of Inn( G ) on T extends to a G -compatible action of Aut( G ) on T . Note that if G splits as a free product A ∗ B , then for any vertex v in T the family of stabilizersof v -incident edges is maximal. Moreover, if A and B are indecomposable, not infinite cyclic andnon isomorphic to each other, then Λ T coincides with Aut( G ) (see [20, p. 152]). As a consequenceof Theorem 14, we obtain the following result: Theorem 18
Let G be a group that splits as a free product A ∗ B with A and B indecomposable,not infinite cyclic and non isomorphic to each other. Then, the induced action of Inn( G ) on T extends to a G -compatible action of Aut( G ) on T . It is natural to ask whether there is a result equivalent to Theorem 18 for a group that splitsas a free product of three or more indecomposable groups.
Question 19 If G splits as the free product of three (or more) indecomposable (one-ended) sub-groups, is there a simplicial tree on which G acts without inversions such that the induced actionof Inn( G ) on that tree can be extended to a finite index subgroup of Aut( G ) ? For some important contributions to the study of the group of automorphisms of a free productof a finite number of indecomposable groups see for instance the work of Foux-Rabinovitch [8, 9].
Interesting examples of groups with a non-trivial splitting are given by one-ended hyperbolic groupswhose group of outer automorphisms is infinite. In Section 2.3 we saw that a surface group splitsover a cyclic subgroup. The last assertion holds as well for a group which is virtually a surfacegroup. When the group is a one-ended hyperbolic group which is not virtually a surface group,the theory of JSJ decompositions guarantees the existence of a non-trivial splitting over a virtuallycyclic subgroup under the assumption that its group of outer automorphisms is infinite (see forinstance Theorem 1.4 in [14]). The notion of JSJ decomposition was introduced in [18] by Sela8or one-ended hyperbolic groups without torsion. In [3], Bowditch constructed a slightly differentJSJ decomposition for one-ended hyperbolic groups (not necessarily torsion-free) by studying cutpoints on the boundary of the group. Both decompositions are “canonical” and they coincide whenthe group is torsion-free. In the rest of this section, G will denote a one-ended hyperbolic groupwithout torsion. Theorem 20 (Sela, Bowditch)
Suppose that
Out ( G ) is an infinite group and that G is notvirtually a surface group. Then there is a non-trivial splitting of G , given by a minimal actionof G on a tree T without inversions and finite quotient T /G . Moreover, the edge stabilizers areall cyclic subgroups and the set of vertex stabilizers is Aut( G ) -invariant. In particular, there is afinite index subgroup of Aut( G ) that preserves this splitting i.e. that preserves the conjugacy classof each vertex stabilizer and each edge stabilizer. This splitting of G is called the JSJ splitting. The adjacency relation on the set of verticesinduces a well-defined binary relation on the set of vertex stabilizers. The JSJ splitting is saidcanonical since Aut( G ) preserves the set of vertex stabilizers with the induced binary relation. Infact, by considering the action of G on ∂G , Bowditch proved in [3] that the extended action ofAut( G ) on T is G -compatible. We refer to [3] and [14] for a detailed description of this splitting.As in the case of a group that splits over a subgroup C , we obtain that the action of G on T isminimal and without fixed points on the boundary. In this section we will prove our main results.
Let Γ be a K¨ahler group that fits in a short exact sequence1 (cid:47) (cid:47) G (cid:47) (cid:47) Γ P (cid:47) (cid:47) Q (cid:47) (cid:47) , (3)where G is a finitely generated group with trivial center acting on a tree T . Suppose that theaction of G on T is minimal, faithful and without fixed points on the boundary. Note that in thiscase the action of G on T coincides with the induced action of Inn( G ) on T .Let Γ be the image of the morphism Γ → Aut( G ) induced by the conjugation action of Γ on G . The main result of this section is the following: Theorem 21
Suppose that there is a finite index subgroup Γ of Γ containing Inn( G ) such thatthe action of G on T can be extended to a G -compatible action of Γ on T . Then G is virtuallya surface group. Moreover, there is a finite index subgroup Γ of Γ containing G such that therestricted short exact sequence (cid:47) (cid:47) G (cid:47) (cid:47) Γ P (cid:47) (cid:47) P (Γ ) (cid:47) (cid:47) splits as a direct product. To prove Theorem 21 we will need a result of Gromov and Schoen [11] on K¨ahler groups actingon trees. For the reader’s convenience we recall some properties of holomorphic maps betweencompact complex manifolds and closed Riemann surfaces. A surjective holomorphic map with9onnected fibers f : X → S from a compact complex manifold X to a closed Riemann surface S induces an orbifold structure as follows. For every point p in S , let m ( p ) be the greatest commondivisor of the multiplicities of the irreducible components of the fiber f − ( p ) and let ∆ be the setof points in S such that m ( p ) ≥
2. Note that ∆ is finite since it is contained in the set of criticalvalues of f . Hence, if ∆ = { p , ..., p k } and m i = m ( p i ) for i = 1 , ..., k , the orbifold structure Σ f isgiven by Σ f = { S ; ( p , m ) , ..., ( p k , m k ) } . For all i = 1 , ..., k let γ i be a loop around p i given by the boundary of a small enough disk suchthat p i is the only singular value contained in the disk. The fundamental group of the orbifold Σ f is given by π orb (Σ f ) = π ( S \ ∆) // (cid:28) γ m , ..., γ m k k (cid:29) , where (cid:28) γ m , ..., γ m k k (cid:29) is the normal closure of { γ , ..., γ k } in π ( S \ ∆). The following lemma isa well-known result. A more general statement of this result can be found in [6]. Lemma 22
Let X be a compact complex manifold, S be a closed Riemann surface and f : X → S be a surjective holomorphic map whose generic fiber F is connected. If we denote by ι : F (cid:44) → X the inclusion map and by N the image of the induced homomorphism on fundamental groups ι ∗ : π ( F ) → π ( X ) , we obtain a short exact sequence (cid:47) (cid:47) N (cid:47) (cid:47) π ( X ) Π (cid:47) (cid:47) π orb (Σ f ) (cid:47) (cid:47) . Since F is compact we get that N is finitely generated. The kernel of the map ρ : π orb (Σ f ) → π ( S ) is isomorphic to ker( f ∗ ) /N and we have the commutative diagram π ( X ) f ∗ (cid:47) (cid:47) Π (cid:37) (cid:37) π ( S ) π orb (Σ f ) ρ (cid:58) (cid:58) To simplify the notation we will abbreviate Σ f to Σ. The result of Gromov and Schoen is thefollowing. Theorem 23 (Gromov-Schoen)
Let X be a compact K¨ahler manifold whose fundamental group Γ acts on a tree non-isomorphic to a line nor a point. Suppose that the action is minimal withno fixed points on the boundary. Then there is a surjective holomorphic map with connected fibersfrom X to a closed hyperbolic surface inducing the short exact sequence (cid:47) (cid:47) N (cid:47) (cid:47) Γ Π (cid:47) (cid:47) π orb (Σ) (cid:47) (cid:47) , such that the restriction of the action to N is trivial. Note that π orb (Σ) is virtually isomorphic to a surface group. Proof of Theorem 21.
Let Γ be the finite index subgroup of Γ satisfying the hypothesis of thetheorem and let Γ be the preimage of Γ under the morphism Γ → Aut( G ). Then, Γ is a finiteindex subgroup of Γ containing G such that the action of G on T can be extended to Γ . ByTheorem 23 there is a short exact sequence1 (cid:47) (cid:47) N (cid:47) (cid:47) Γ (cid:47) (cid:47) π orb (Σ) (cid:47) (cid:47) , on T to the subgroup N is trivial and π orb (Σ) is virtuallyisomorphic to a surface group. Now, let λ : G → π orb (Σ) be the morphism given by the restrictionof Π to G . This morphism is injective since N ∩ G is trivial. The latter assertion follows from thefaithfulness of the action of G on T . We claim that the subgroup Γ = Π − ( λ ( G )) is the subgroupwe are looking for. First of all, since for a normal subgroup of π orb (Σ) being finitely generated isequivalent to having finite index, we get that λ ( G ) has finite index in π orb (Σ). This implies that Γ is a finite index subgroup of Γ (and thereby of Γ) and that G is virtually isomorphic to a surfacegroup. To end the proof, it suffices to show that the morphism η : Γ → G × P (Γ ) x (cid:55)→ ( λ − (Π( x )) , P ( x ))is bijective. The injectivity follows from the fact that the restriction of λ − ◦ Π to G is the identity.For the surjectivity, if ( x , q ) is an element of G × P (Γ ), by taking y in Γ such that P ( y ) = q and x in G such that λ ( x ) = Π( y ), we get that η ( x x − y ) = ( x , q ). (cid:50) We used the fact that if the action of G on T is minimal without fixed points on the boundary,then any extension of this action has these properties as well. Here we will apply Theorem 21 and the results of Section 2 to prove our main results. The proofof Theorem A will require the following result:
Proposition 24
Let S be a closed hyperbolic surface and let C be the cyclic subgroup of π ( S ) generated by the homotopy class of a simple closed curve γ . Then the action of π ( S ) on theBass-Serre tree associated to the splitting of π ( S ) over C is faithful.Proof. Let T be the Bass-Serre tree associated to the splitting of π ( S ) over C . Recall that T coincides with the dual tree associated to γ (see Remark 9). This identification is given by abijection between the edges of T and a set of disjoint bi-infinite geodesics in H . Hence, an edgestabilizer of T coincides with the stabilizer of a bi-infinite geodesic in H under the action of π ( S )on H . From this, we conclude that the intersection of the stabilizers of two different edges of T istrivial, which implies the faithfulness of the action of G on T . (cid:50) Proof of Theorem A.
Let γ be the simple closed curve in S , whose conjugacy class is preservedby the conjugation action of Γ on π ( S ). Let C be the cyclic subgroup of π ( S ) generated bythe homotopy class of γ and let T be the Bass-Serre tree associated to the splitting of π ( S )over C . Therefore, the image of Γ → Aut( π ( S )) denoted by Γ is contained in the subgroup ofautomorphisms of π ( S ) that preserves the conjugacy class of γ . By Proposition 11, the action of π ( S ) on T can be extended to a π ( S )-compatible action of a subgroup of Γ of index at most 2on T . By Proposition 24 we have that the action of π ( S ) on T is faithful and the result followsfrom Theorem 21. (cid:50) Let G be a group that splits over a subgroup C and let T be its Bass-Serre tree. Suppose thatfor an edge of T , its stabilizer is properly contained in its vertex stabilizers. Suppose as well thatfor any vertex v of T , the family of stabilizers of v -incident edges is maximal. Let Γ be a K¨ahlergroup containing G as a normal subgroup such that the conjugation action of Γ on G preserves theconjugacy classes of edge stabilizers and vertex stabilizers of T . If the action of G on its Bass-Serretree is faithful, then as a consequence of Theorems 14, 15 and 21 we have that G must be virtuallya surface group. Hence, Theorem B follows from the latter remark and the following proposition.11 roposition 25 If a surface group is embedded in a free product A ∗ B , then it is embedded in A or it is embedded in B . In particular, a non-trivial free product of two groups is not virtually asurface group.Proof. Let A and B be groups with at least two elements. Since A and B are both infinite indexsubgroups of A ∗ B , the second part of the proposition follows from the first one. First, we willprove that a non-trivial free product is not isomorphic to a surface group. Let us assume bycontradiction that the free product A ∗ B is isomorphic to a surface group. Recall that an infiniteindex subgroup of a surface group is a free group. Hence, since A and B are both infinite indexsubgroups of A ∗ B , we obtain that they are both free groups. Thus, the free product A ∗ B is alsoa free group, which is a contradiction. Note that the same argument shows that a surface group isnot isomorphic to a free product with more than two free factors. The general case follows fromKurosh subgroup Theorem (see [20, p. 151]), which states that a subgroup of the free product A ∗ B is given by the free product of a free group with subgroups of conjugates of A and B . (cid:50) Theorem C follows from Theorem 17, the remark that precedes Proposition 25 and the twofollowing propositions:
Proposition 26
The Baumslag-Solitar group G ( p, q ) = (cid:104) x, t | tx p t − = x q (cid:105) , where p, q are different integers with p, q > , is not virtually a surface group.Proof. If | p | (cid:54) = | q | and if we suppose that G ( p, q ) is a surface group, we would have that G ( p, q ) actson H . Hence, if we denote by (cid:96) the length translation of x in H , as a consequence of the relation tx p t − = x q , we would have that p(cid:96) = q(cid:96) which is impossible. Since any non-cyclic subgroup of G ( p, q ) is free or contains conjugated subgroups of (cid:104) x p (cid:105) and (cid:104) x q (cid:105) , the same argument as beforeshows that G ( p, q ) is not virtually a surface group.If p = q , the center of the group G ( p, p ) is infinite and it contains x p . Hence the center of anyfinite index subgroup of G ( p, p ) must be infinite. This implies that no such finite index subgroupis a surface group and therefore G ( p, p ) is not virtually a surface group.Finally, by considering the morphisms from G ( p, p ) and G ( p, − p ) to Z / Z that send x to theidentity and t to the generator of Z / Z , we obtain that both groups contain as a subgroup of index2 the group given by the presentation (cid:104) x, y, u | x p y p = 1 , [ u, x p ] = 1 (cid:105) . Therefore, G ( p, p ) and G ( p, − p ) are commensurable and since G ( p, p ) is not virtually a surfacegroup we obtain that neither is G ( p, − p ). (cid:50) Proposition 27
The Baumslag-Solitar group G ( p, q ) = (cid:104) x, t | tx p t − = x q (cid:105) , where p, q are different integers with p, q > , acts faithfully on its Bass-Serre tree.Proof. The kernel N of the action of G ( p, q ) on its Bass-Serre tree is contained in every edgestabilizer. In particular, it is contained in (cid:104) x p (cid:105) . Assume by contradiction that N is non-trivial.Then, N is generated by x kp for some nonzero integer k . Since N is a normal subgroup of G ( p, q ),we get that tx kp t − = x kq is an element of (cid:104) x kp (cid:105) . Therefore kq = mkp and p divides q . Bysymmetry, we obtain that q divides p and thereby p = q , which is a contradiction. (cid:50) Here we present a different proof of Bregman and Zhang’s result using Theorem A.
Corollary 28 (Bregman-Zhang)
Let S be a closed hyperbolic surface, Γ a K¨ahler group and k a positive integer such that there is a short exact sequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) Z k (cid:47) (cid:47) . Then there is a finite index subgroup Γ of Γ containing π ( S ) such that the restricted short exactsequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) P (Γ ) ∼ = Z k (cid:47) (cid:47) splits as a direct product. Note that k must be even since Γ is a K¨ahler group. Before proving Corollary 28, we willrecall some definitions and facts about the mapping class group of a closed hyperbolic surface (thiscan be found with more details in [2]). Let S ( S ) denote the set of isotopy classes of simple closedcurves in S . If τ is an element of the mapping class group of S and A is a subset of S ( S ), wedenote by τ ( A ) the set { τ ( α ) | α ∈ A } , where τ ( α ) denotes the isotopy class of t ( a ) for t ∈ τ and a ∈ α . A subset A of S ( S ) is saidadmissible if there is a set of disjoint simple closed curves that represent all the isotopy classes of A . An element τ of the mapping class group of S is said to be1. Reducible . If there is a non-empty admissible set A such that τ ( A ) = A .2. Pseudo-Anosov . If for any isotopy class α in S ( S ) and for all non-zero integer n , τ n ( α )is different from α (this is one definition of a pseudo-Anosov mapping class among manyothers).To prove Corollary 28 we will need the classification theorem of Thurston and two Lemmas. Thefirst Lemma is due to Birman, Lubotzky and McCarthy (see Lemma 3.1 in [2]). Lemma 29 (Birman-Lubotzky-McCarthy)
Let A be an abelian subgroup of the mapping classgroup of a hyperbolic surface generated by reducible elements { τ , ..., τ k } . Then, there is a non-empty canonical admissible set A such that τ i ( A ) = A for all i = 1 , ..., k . The second Lemma is the following. It will be a consequence of Thurston’s hyperbolizationTheorem (see [16]), together with a classical result of Carlson and Toledo [5].13 emma 30
Let S be a hyperbolic surface, Γ a K¨ahler group and k a positive integer such thatthere is a short exact sequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) Z k (cid:47) (cid:47) . Then the image of the monodromy map ψ : Z k → M CG ( S ) does not contain pseudo-Anosovelements.Proof. Let us assume by contradiction that the image of ψ contains a pseudo-Anosov element τ . It is well known that the cyclic subgroup generated by a pseudo-Anosov element is a finiteindex subgroup of its centralizer (see [15]). Hence, by passing to a finite index subgroup of Γ ifnecessary (which is also a K¨ahler group), we may assume that the image of ψ is the cyclic subgroupgenerated by τ . Let { e , ..., e k } be a basis of Z k such that { e , ..., e k − } generates the kernel of ψ and ψ ( e k ) = τ . We denote by Γ the subgroup of Γ generated by π ( S ) and P − ( e k ). Claim 31 Γ is isomorphic to Γ × Z k − . Claim 31 implies the result since by Thurston’s hyperbolization Theorem (see [16]), Γ is a co-compact lattice in the group of orientation preserving isometries of the hyperbolic 3-space and byCarlson and Toledo’s Theorem (see [5]), the projection of Γ onto the cocompact lattice Γ factorsthrough a surface group Λ. Hence, there is a commutative diagramΓ ∼ = Γ × Z k P (cid:47) (cid:47) θ (cid:37) (cid:37) Γ Λ (cid:63) (cid:63) where P is the projection on Γ . This leads to the desired contradiction, since θ (Γ ) is a subgroupof Λ isomorphic to Γ (as a consequence of the fact that P | Γ is the identity map), and everysubgroup of Λ is the fundamental group of a (closed or open) surface, but Γ is neither free norisomorphic to the fundamental group of a closed oriented surface since H (Γ , Z ) is non-trivial. (cid:50) Proof of Claim 31.
Observe that for each e i in the kernel of ψ , there is a unique element g i inΓ such that g i centralizes π ( S ) and P ( g i ) = e i . Since the commutator [ g i , g j ] is an element of π ( S ) which centralizes π ( S ) we get that the group K generated by { g , ..., g k − } is isomorphic to Z k − and by construction [ K, π ( S )] is trivial. Using the fact that π ( S ) is a normal subgroup ofΓ and that K centralizes π ( S ) we obtain that [ K, Γ] centralizes π ( S ). Finally, since [ K, Γ] is inthe kernel of P (which is π ( S )) we conclude that [ K, Γ] is in the center of π ( S ) which is trivial.Therefore the subgroups K and Γ commute and the result follows from observing that Γ = K Γ and K ∩ Γ is trivial. (cid:50) Now, we give our proof of Corollary 28:
Proof of Corollary 28.
By Thurston’s classification Theorem and by Lemma 30, up to passing to afinite index subgroup of Z k we have that the image of the monodromy map ψ : Z k → M CG ( S ) isgenerated by reducible elements. Hence, by Lemma 29 there is a non-empty canonical admissibleset A of S ( S ) such that ψ ( Z k ) preserves A . Since a finite index subgroup of Z k (which isisomorphic to Z k ) acts trivially on A , we can apply Theorem A which concludes the proof. (cid:50) With a bit more work we get the next corollary of Theorem A which is more general than Corollary28. 14 orollary 32
Let S be a hyperbolic surface and Γ be a K¨ahler group such that there is a shortexact sequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) Q (cid:47) (cid:47) . If the image of the monodromy map ψ : Q → M CG ( S ) is abelian, then there are finite indexsubgroups Γ of Γ and Q of Q such that π ( S ) is contained in Γ and the restricted short exactsequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ (cid:47) (cid:47) Q (cid:47) (cid:47) splits as a direct product. To prove Corollary 32 it suffices to prove a result analogous to Lemma 30 for the case whenthe monodromy map ψ : Q → M CG ( S ) has abelian image (instead of assuming that Q itself isabelian). According to a result of Birman, Lubotzky and McCarthy (see [2] Theorem B),everysolvable subgroup of the mapping class group of a hyperbolic surface is virtually abelian. Hence,Corollary 32 can be extended to the case when ψ ( Q ) is a solvable group. Lemma 33
Let S be a hyperbolic surface and Γ be a K¨ahler group such that there is a short exactsequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) Q (cid:47) (cid:47) . If the image of the monodromy map ψ : Q → M CG ( S ) is abelian, then it does not contain pseudo-Anosov elements.Proof. The proof is by contradiction. Let us suppose that the image of ψ contains a pseudo-Anosovelement τ . As in Lemma 30, up to passing to a finite index subgroup of Γ we may assume that theimage of ψ is the cyclic subgroup generated by τ . Let t be an element in Γ such that ψ ◦ P ( t ) = τ and let Γ be the subgroup of Γ generated by π ( S ) and t . The result will follow from exhibitinga homomorphism Γ → Γ whose restriction to Γ is the identity, since Thurston’s hyperbolizationTheorem and Carlson and Toledo’s Theorem will lead us to a contradiction as in Lemma 30.Now, to construct this homomorphism, observe that since each element in the kernel of ψ has aunique lift in Γ that centralizes π ( S ), there is a subgroup K of Γ isomorphic to the kernel of ψ thatcentralizes π ( S ). Indeed, K is the centralizer of π ( S ) in Γ. Hence, the subgroup [Γ , K ] centralizes π ( S ) and since its image under P is contained in the kernel of ψ , we get that [Γ , K ] is containedin K and therefore K is a normal subgroup of Γ. Finally, since the kernel of ψ ◦ P is isomorphicto K × π ( S ), we obtain that Γ is isomorphic to the semi-direct product ( K × π ( S )) (cid:111) (cid:104) t (cid:105) , andthereby there is a well-defined homomorphismΓ → Γ kxt m (cid:55)→ xt m , where k is in K , x is in π ( S ) and m is in Z . To verify that it is a well-defined map we just needto observe that for k x t m and k x t m in Γ we have that k x t m k x t m = ( k t m k t − m )( x t m x t − m ) t m + m , where we notice that t m k t − m is in K and t m x t − m is in π ( S ). (cid:50) The following result is a consequence of a Theorem due to Ivanov which states the following. Let S be a closed hyperbolic surface and Γ an infinite subgroup of M CG ( S ) that does not preserve anyadmissible set A of S ( S ). Then, Γ is either virtually a cyclic group generated by a pseudo-Anosovelement, or it contains a free group generated by 2 pseudo-Anosov elements (see [12] Theorem 2).15 orollary 34 Let S be a hyperbolic surface and Γ a a K¨ahler group such that there is a shortexact sequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) Q (cid:47) (cid:47) . If the image of the monodromy map ψ : Q → M CG ( S ) is infinite, then it contains a free subgroupgenerated by pseudo-Anosov elements.Proof. Recall that an extension as above is virtually trivial if and only if the monodromy subgroupof the short exact sequence is finite. Therefore, by Theorem A ψ ( Q ) does not preserve any admis-sible set A of S ( S ). Finally, the result follows from Lemma 33 and Ivanov’s result recalled beforethe statement. (cid:50) A classical result due to Scott (see [19]) allows us to extend Theorem A to any closed curve in S . Theorem 35 (Scott)
Let S be a topological surface and let x be an element of π ( S ) . Then,there is a finite covering map S (cid:48) → S such that x is in π ( S (cid:48) ) and can be represented by a simpleclosed curve in S (cid:48) . Lemma 36
Let G be a group and H be a finitely generated normal subgroup of G . If Λ is a finiteindex subgroup of H , then the normalizer of Λ in G is a finite index subgroup of G .Proof. Let n be the index of Λ in H . Then G acts on the set of subgroups of H of index n byconjugation. This set is finite and therefore the stabilizer of any of these subgroups gives a finiteindex subgroup of G . Finally, notice that the stabilizer of Λ is precisely the normalizer of Λ in G and the result follows. (cid:50) Theorem 37
Let Γ be a K¨ahler group such that there is a short exact sequence (cid:47) (cid:47) π ( S ) (cid:47) (cid:47) Γ P (cid:47) (cid:47) Q (cid:47) (cid:47) , with S a closed hyperbolic surface. If the conjugation action of Γ on π ( S ) preserves the conjugacyclass of a non-trivial element of π ( S ) , then there is a finite index subgroup Γ of Γ containing afinite index subgroup Λ of π ( S ) which is normal in Γ such that the extension (cid:47) (cid:47) Λ (cid:47) (cid:47) Γ (cid:47) (cid:47) Γ / Λ (cid:47) (cid:47) splits as a direct product.Proof. Let γ be the closed curve in S whose conjugacy class is preserved by the action of Γ. Wemay assume that γ is not simple. By Theorem 35, there is a finite covering map S (cid:48) → S , suchthat γ lifts to a simple closed curve in S (cid:48) . Let Λ be the fundamental group of S (cid:48) and let Γ (cid:48) be thenormalizer of Λ in Γ. By Lemma 36, Γ (cid:48) is a finite index subgroup of Γ and therefore Γ (cid:48) is a K¨ahlergroup. Hence, the short exact sequence1 (cid:47) (cid:47) Λ (cid:47) (cid:47) Γ (cid:48) (cid:47) (cid:47) Γ (cid:48) / Λ (cid:47) (cid:47) of Γ (cid:48) of finiteindex (and therefore a finite index subgroup of Γ) such that the exact sequence1 (cid:47) (cid:47) Λ (cid:47) (cid:47) Γ (cid:47) (cid:47) Γ / Λ (cid:47) (cid:47) , (cid:50) Acknowledgements.
This article is a part of my PhD thesis. I would like to thank my PhDadvisors, Thomas Delzant and Pierre Py who proposed this subject to me, for all the discussionsconcerning this work and for their great support. I would also like to thank Martin Mion-Moutonand Ad`ele P´erus for their comments on this text.
References [1] J. Amor´os, M. Burger, K. Corlette, D. Kotschick and D. Toledo
Fundamental groups of com-pact K¨ahler manifolds , Mathematical Surveys and Monographs. , American MathematicalSociety, Providence, RI, (1996). – cited on p. 2[2] J.S. Birman, A. Lubotzky and J. McCarthy Abelian and solvable subgroups of the mappingclass groups , Duke Mathematical Journal. , No. 4, (1983), 1107–1120. – cited on p. 13, 15[3] B.H. Bowditch Cut points and canonical splittings of hyperbolic groups , Acta Mathemat-ica. , No. 2, (1998), 145–186. – cited on p. 1, 2, 9[4] C. Bregman and L. Zhang
On K¨ahler extensions of abelian groups , arXiv e-prints, (2016). –cited on p. 2[5] J.A. Carlson and D. Toledo
Harmonic mappings of K¨ahler manifolds to locally symmetricspaces , Institut des Hautes ´Etudes Scientifiques. Publications Math´ematiques, No. 69, (1989),173–201. – cited on p. 13, 14[6] F. Catanese, J. Keum and K. Oguiso
Some remarks on the universal cover of an open K surface , Mathematische Annalen, , No. 2, (2003), 279–286. – cited on p. 10[7] B. Farb and D. Margalit A primer on mapping class groups , Princeton Mathematical Se-ries. , Princeton University Press, Princeton, NJ, (2012). – cited on p. 7[8] D.I. Fouxe-Rabinovitch ¨Uber die Automorphismengruppen der freien Produkte. I , Rec. Math.[Mat. Sbornik] N.S., , (1940), 265–276. – cited on p. 8[9] D.I. Fouxe-Rabinovitch ¨Uber die Automorphismengruppen der freien Produkte. II , Rec. Math.[Mat. Sbornik] N. S., , (1941), 183–220. – cited on p. 8[10] N.D. Gilbert, J. Howie, V. Metaftsis and E. Raptis Tree actions of automorphism groups ,Journal of Group Theory. , No. 2, (2000), 213–223. – cited on p. 1, 2[11] M. Gromov and R. Schoen Harmonic maps into singular spaces and p -adic superrigidityfor lattices in groups of rank one , Institut des Hautes ´Etudes Scientifiques. PublicationsMath´ematiques, No. 76, (1992), 165–246. – cited on p. 2, 9[12] N.V. Ivanov Subgroups of Teichm¨uller modular groups , Translations of Mathematical Mono-graphs. , Translated from the Russian by E. J. F. Primrose and revised by the author,American Mathematical Society, Providence, RI, (1992). – cited on p. 15[13] A. Karrass, A. Pietrowski and D. Solitar
Automorphisms of a free product with an amalgamatedsubgroup , Contributions to group theory, Contemp. Math., Amer. Math. Soc., Providence,RI. , (1984), 328–340. – cited on p. 1, 2, 7[14] G. Levitt Automorphisms of hyperbolic groups and graphs of groups , Geometriae Dedi-cata. , (2005), 49–70. – cited on p. 8, 91715] J. McCarthy
Normalizers and centralizers of pseudo-Anosov mapping classes ,https://users.math.msu.edu/users/mccarthy/publications/normcent.pdf. – cited on p. 14[16] J.P. Otal
Le th´eor`eme d’hyperbolisation pour les vari´et´es fibr´ees de dimension 3 , Ast´erisque,No. 235, (1996). – cited on p. 13, 14[17] M.R. Pettet
The automorphism group of a graph product of groups , Communications in Alge-bra. , No. 10, (1999), 691–4708. – cited on p. 1, 2, 8[18] Z. Sela Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank Liegroups. II , Geometric and Functional Analysis. , No. 3, (1997), 561–593. – cited on p. 1, 2,8[19] P. Scott Subgroups of surface groups are almost geometric , Journal of the London Mathemat-ical Society. Second Series , No. 3, (1978), 555–565. – cited on p. 16[20] P. Scott and T. Wall Topological methods in group theory , Homological group theory (Proc.Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press,Cambridge-New York. , (1979), 137–203. – cited on p. 1, 3, 8, 12[21] J.P. Serre Trees , Springer Monographs in Mathematics, Translated from the French originalby John Stillwell, Corrected 2nd printing of the 1980 English translation, Springer-Verlag,Berlin, (2003). – cited on p. 1, 3[22] H. Shiga
On monodromies of holomorphic families of Riemann surfaces and modular trans-formations , Mathematical Proceedings of the Cambridge Philosophical Society. , No. 3,(1997), 541–549. – cited on p. 2