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Nielsen Realisation by Gluing: Limit Groups and Free Products

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NIELSEN REALISATION BY GLUING: LIMIT GROUPS AND FREE PRODUCTS SEBASTIAN HENSEL AND DAWID KIELAK Abstract. We generalise the Karrass{Pietrowski{Solitar and the Nielsen realisation theorems from the setting of free groups to that of free prod- ucts. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel{Mosher and on the outer space of a free product of Guirardel{ Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained. 1. Introduction In its original form, the Nielsen realisation problem asks which finite sub- groups of the mapping class group of a surface can be realised as groups of homeomorphisms of the surface. A celebrated result of Kerckhoff [Ker1, Ker2] answers this in the positive for all finite subgroups, and even allows for realisations by isometries of a suitable hyperbolic metric. Subsequently, similar realisation results were found in other contexts, per- haps most notably for realising finite groups in Out(Fn) by isometries of a suitable graph (independently by [Cul], [Khr], [Zim]; compare [HOP] for a different approach). In this article, we begin to develop a relative approach to Nielsen realisa- tion problems. The philosophy here is that if a group G allows for a natural arXiv:1601.02187v3 [math.GR] 14 Feb 2018 decomposition into pieces, then Nielsen realisation for Out(G) may be re- duced to realisation in the pieces, and a gluing problem. In addition to just solving Nielsen realisation for finite subgroups of Out(G), such an approach yields more explicit realisations, which also exhibit the structure of pieces for G. We demonstrate this strategy for two classes of groups: free products and limit groups. In another article, we use the results presented here, together with the philosophy of relative Nielsen realisation, to prove Nielsen realisation for certain right-angled Artin groups ([HK]). The second-named author was supported by the SFB 701. 1 2 SEBASTIAN HENSEL AND DAWID KIELAK The early proofs of Nielsen realisation for free groups rely in a fundamental way on a result of Karrass{Pietrowski{Solitar [KPS], which states that every finitely generated virtually free group acts on a tree with finite edge and vertex stabilisers. In the language of Bass{Serre theory, it amounts to saying that such a virtually free group is a fundamental group of a finite graph of groups with finite edge and vertex groups. This result of Karrass{Pietrowski{Solitar in turn relies on the celebrated theorem of Stallings on groups with at least two ends [Sta1, Sta2]. Stallings' theorem states that any finitely generated group with at least two ends splits over a finite group, which means that it acts on a tree with a single edge orbit and finite edge stabilisers. Equivalently: it is a fundamental group of a graph of groups with a single edge and a finite edge group. In the first part of this article, we generalise these results to the setting of a free product A = A1 ∗ · · · ∗ An ∗ B in which we (usually) require the factors Ai to be finitely generated, and B to be a finitely generated free group. Consider any finite group H acting on A by outer automorphisms in a way preserving the given free-product decomposition, by which we mean that each element of H sends each sub- group Ai to some Aj (up to conjugation); note that we do not require the action of H to preserve B in any way. We then obtain a corresponding group extension 1 ! A ! A ! H ! 1 In this setting we prove (for formal statements, see the appropriate sections) Relative Stallings' theorem (Theorem 2.7): A splits over a finite group, in such a way that each Ai fixes a vertex in the associated action on a tree. Relative Karrass{Pietrowski{Solitar theorem (Theorem 4.1): A acts on a tree with finite edge stabilisers, and with each Ai fixing a vertex of the tree, and with, informally speaking, all other vertex groups finite. Relative Nielsen realisation theorem (Theorem 7.5): Suppose that we are given complete non-positively curved (i.e. locally CAT(0)) spaces Xi realising the induced actions of H on the factors Ai. Then the action of H can be realised by a complete non-positively curved space X; in fact X can be chosen to contain the Xi in an equivariant manner. We emphasise that such a relative Nielsen realisation is new even if all Ai are free groups, in which case it answers a question of Karen Vogtmann. The classical Nielsen realisation for graphs immediately implies that a finite subgroup H < Out(Fn) fixes points in the Culler{Vogtmann Outer Space (defined in [CV]), as well as in the complex of free splittings of Fn (which is a simplicial closure of Outer Space). As an application of the NIELSEN REALISATION BY GLUING: LIMIT GROUPS AND FREE PRODUCTS 3 work in this article, we similarly obtain fixed point statements (Corollar- ies 5.1 and 6.1) for the graph of relative free splittings defined by Handel and Mosher [HM], and the outer space of a free product defined by Guirardel and Levitt [GL]. In the last section of the paper we prove Theorem 8.12. Let A be a limit group, and let A ! A ! H be an extension of A by a finite group H. Then there exists a complete compact locally CAT(κ) space X realising the extension A, where κ = −1 when A is hyperbolic, and κ = 0 otherwise. When κ = −1, the space X is of dimension at most 2. This theorem is obtained by combining the classical Nielsen realisation theorems (for free, free-abelian and surface groups { see Theorems 8.2 to 8.4) with the existence of an invariant JSJ decomposition shown by Bumagin{ Kharlampovich{Myasnikov [BKM]. Note that, in general, having a graph of groups decomposition for a group G with CAT(0) vertex groups and virtually cyclic edge groups does not allow one to build a CAT(0) space for G to act on, and thus conclude that G is itself CAT(0); the JSJ decompositions of limit groups are however special in this respect, and the extra structure allows for the conclusion. This has been observed by Sam Brown in [Bro], where he developed techniques for building up a CAT(0) space for G to act on. Observe that we obtain optimal curvature bounds for our space X { it has been proved by Alibegovi´c{Bestvina[AB] that limit groups are CAT(0), and by Sam Brown [Bro] that a limit group is CAT(−1) if and only if it is hyperbolic. Also, taking H to be the trivial group gives a new (more direct) proof of the fact that limit groups are CAT(0). Throughout the paper, we are going to make liberal use of the standard terminology of graphs of groups. The reader may find all the necessary information in Serre's book [Ser]. We are also going to make use of standard facts about CAT(0) and non-positively curved (NPC) spaces, as well as more general CAT(κ) spaces; the standard reference here is the book by Bridson{ Haefliger [BH]. Acknowledgements. The authors would like to thank Karen Vogtmann for discussions and suggesting the statement of relative Nielsen realisation for free groups, Stefan Witzel for pointing out the work of Sam Brown, and the referee for extremely valuable comments. 2. Relative Stallings' theorem In this section we will prove the relative version of Stallings' theorem. Before we can begin with the proof, we need a number of definitions to 4 SEBASTIAN HENSEL AND DAWID KIELAK formalise the notion of a free splitting that is preserved by a finite group action. Convention. When talking about free factor decompositions A = A1 ∗ · · · ∗ An ∗ B of some group A, we will always assume that at least two of the factors fA1;:::;An;Bg are non-trivial. Definition 2.1. Suppose that φ: H ! Out(A) is a homomorphism with a finite domain. Let A = A1 ∗ · · · ∗ An ∗ B be a free factor decomposition of A. We say that this decomposition is preserved by H if and only if for every i and every h 2 H, there is some j such that h(Ai) is conjugate to Aj. We say that a factor Ai is minimal if and only if for any h 2 H the fact that h(Ai) is conjugate to Aj implies that j > i. Remark 2.2. Note that when the decomposition is preserved, we obtain an induced action H ! Sym(n) on the indices 1; : : : ; n. We may thus speak of the stabilisers StabH (i) inside H. Furthermore, we obtain an induced action StabH (i) ! Out(Ai) The minimality of factors is merely a way of choosing a representative of each H orbit in the action H ! Sym(n). Remark 2.3. Given an action φ: H ! Out(A), with φ injective and A with trivial centre, we can define A 6 Aut(A) to be the preimage of H = im φ under the natural map Aut(A) ! Out(A). We then note that A is an extension of A by H: 1 ! A ! A ! H ! 1 and the left action of H by outer automorphisms agrees with the left con- jugation action inside the extension A.
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