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Max-Planck-Institut Für Mathematik Bonn Max-Planck-Institut für Mathematik Bonn Similar relatively hyperbolic actions of a group by Victor Gerasimov Leonid Potyagailo Max-Planck-Institut für Mathematik Preprint Series 2014 (34) Similar relatively hyperbolic actions of a group Victor Gerasimov Leonid Potyagailo Max-Planck-Institut für Mathematik Departamento de Matemática Vivatsgasse 7 Universidade Federal de Minas Gerais 53111 Bonn Av. Antonio Carlos 6627 Germany Caixa Postal 702 CEP 31270-901 Belo Horizonte Brasil UFR de Mathématiques Université de Lille 1 59655 Villeneuve d’Ascq cedex France MPIM 14-34 SIMILAR RELATIVELY HYPERBOLIC ACTIONS OF A GROUP VICTOR GERASIMOV AND LEONID POTYAGAILO Abstract. Let a discrete group G possess two convergence actions by homeomorphisms on compacta X and Y . Consider the following question: does there exist a convergence action GyZ on a compactum Z and continuous equivariant maps X Z ! Y ? We call the space Z (and action of G on it) pullback space (action). In such general setting a negative answer follows from a recent result of O. Baker and T. Riley [BR]. Suppose, in addition, that the initial actions are relatively hyperbolic that is they are non-parabolic and the induced action on the distinct pairs are cocompact. Then the existence of the pullback space if G is finitely generated follows from [Ge2]. The main result of the paper claims that the pullback space exists if and only if the maximal parabolic subgroups of one of the actions are dynamically quasiconvex for the other one. We provide an example of two relatively hyperbolic actions of the free group G of countable rank for which the pullback action does not exist. We study an analog of the notion of geodesic flow for relatively hyperbolic groups. Further these results are used to prove the main theorem. 1. Introduction This paper is a further development of our project of studying convergence group actions including the actions of relatively hyperbolic groups. An action of a discrete group G by homeomorphisms of a compactum X is said to have convergence property if the induced action on the space of distinct triples of X is properly discontinuous. We call such an action 3-discontinuous. The complement ΛX G of the maximal open subset where the action is properly discontinuous is called the limit set of the action. The action is said to be minimal if ΛX G=X. The goal of the paper is to establish similarity properties between different convergence actions of a fixed group. The first motivation for us was the following question: Q1 : Given two minimal 3-discontinuous actions of a group G on compacta X; Y does there exists a 3- discontinuous action G y Z on a compactum Z and continuous equivariant maps X Z ! Y ? We call such an action pullback action and the space Z pullback space. The answer to this question is negative in general. This follows from a recent result of O. Baker and T. Riley [BR]. They indicated a hyperbolic group G and a free subgroup H of rank three such that the embedding H ! G does not admit an equivariant continuous extension to the hyperbolic boundaries @1H ! @1G (so called Cannon- y y Thurston map). It is an easy consequence of this result that 3-discontinuous actions H @1H and H Λ@1GH do not possess a pullback 3-discontinuous action (see section 4). The question Q1 has a natural modification. Suppose in addition that our actions GyX, GyY are 2-cocompact, that is the quotient of the space of distinct pairs of the corresponding space by G is compact. This condition is natural because the class of groups which admit non-trivial 3-discontinuous and 2-cocompact actions (we say 32-actions) coincides with the class of relatively hyperbolic groups [GePo2, Theorem 3.1]. So the following question is the main subject of the paper. Q2 : Given two minimal 32-actions of a group G on compacta X; Y does there exist a 3-discontinuous action GyZ possessing continuous equivariant maps X Z ! Y ? We note that if such an action G y Z exists then one can choose it to be 2-cocompact (see Lemma 5.4). So the pullback action is also of type (32). Recall few standard definitions. An action on a compactum is called parabolic if it admits a unique fixed point. For a 3-discontinuous action G y X a point p 2 X is called parabolic if it is the unique fixed point for its stabilizer Date: May 27, 2014. 2010 Mathematics Subject Classification. Primary 20F65, 20F67; Secondary 57M07, 22D05. Key words and phrases. Convergence actions, relatively hyperbolic group, equivariant map between actions, pullback space. 1 2 VICTOR GERASIMOV AND LEONID POTYAGAILO P = StGp and StGp is infinite. The subgroup P is called a maximal parabolic subgroup. We denote by ParX the set of the parabolic points for an action on X: If G y X is a 32-action then the set of all maximal parabolic subgroups is called peripheral structure for the action. It consists of finitely many conjugacy classes of maximal parabolic subgroups [Ge1, Main theorem, a]. If G is finitely generated then an affirmative answer to the question Q2 can be easily deduced from [Ge2, Map theorem] (see section 5 below). Furthermore there exists a "universal" pullback space in this case. Namely every 32-action of a finitely generated group G on a compactum X admits an equivariant continuous map from the Floyd boundary @f G of G to X. The space @f G is universal as it does not depend on the action on X (it depends on a scalar function f rescaling the word metric of the Cayley graph and a fixed finite set of generators of G). However the same method does not work if the group is not finitely generated. One cannot use the Cayley graph since the quotient of the set of its edges by the group is not finite (the action is not cofinite on edges) the condition which is needed for the construction of the above map. Replacing the Cayley graph by a relative Cayley graph changes the situation since the latter graph depends on the 32-action of G on a compactum X. Indeed the vertex set of the graph contains the parabolic points for the action G y X: This problem turns out to be crucial since the answer to the question Q2 is negative in general. We show in the following theorem that a counter-example exists already in the case of free groups of countable rank. Theorem I (Proposition 5.5). The free group F1 of countable rank admits two 32-actions not having a pullback space. We note that this is a rare example when certain properties of the relatively hyperbolic groups are true for finitely generated groups and are false for non-finitely generated (even countable) groups. Our next goal is to provide necessary and sufficient conditions for two 32-actions of a group to have a common pullback space. The following theorem is the main result of the paper. Theorem II (Theorem 5.6, Theorem B). Two 32-actions of G on compacta X and Y with peripheral structures P and Q admit a pullback space Z if and only if one of the following conditions is satisfied: 1) C(X; Y ): every element P 2 P acts 2-cocompactly on its limit set in Y . 2) C(Y; X): every element Q 2 Q acts 2-cocompactly on its limit set in X. Here are several remarks about the theorem. As an immediate corollary we obtain that C(X; Y ) is equivalent to C(Y; X): This statement seems to be new even in the finitely generated case. It follows from Theorem II that a parabolic subgroup H for a 32-action of a finitely generated group G acts 2-cocompactly on its limit set for every other such action of G: Theorem I implies that this is not true if G is not finitely generated (see Corollary 6.1.f). The peripheral structure R for the pullback action on Z is given by the system of subgroups R = fQ \ P : P 2 P;Q 2 Q; jP \ Qj = 1g: In particular Theorem II provides a criterion when the system R is a peripheral structure for some relatively hyperbolic action of G (Corollary 6.1.a). The proof of Theorem II uses several intermediate results which occupy first sections of the paper and which have independent interest. We will now briefly describe them. In section 3 we study an analog of the geodesic flow introduced by M. Gromov in the case of hyperbolic groups. If the group G admits a 32-action on a compactum X; then there exists a connected graph Γ such that G acts 1 0 properly and cofinitely on the set of edges Γ of Γ [GePo2, Theorem A]. The set of vertices Γ of Γ is ParX t G: The union Xe = X [ Γ0 = X t G admits a Hausdorff topology whose restriction on X and on G coincide with the initial topology and the discrete topology respectively, and G acts on Xe 3-discontinuously [Ge2, Proposition 8.3.1]. The action is also 2-cocompact (Lemma 5.4). We call the space Xe attractor sum of X and G. Consider the space of maps γ : Z ! Xe for which there exist m; n 2 Z [ {±∞} such that γ is constant on one or both (possibly empty) sets ] − 1; m]; [n; +1[ and is geodesic in Γ0 outside of these sets. We call such a map eventual geodesic and denote by EG(Γ) the space of all eventual geodesics.
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