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Relative hyperbolicity of suspensions of free products

Abstract : In this thesis, we are interested in the study of the relative hyperbolicity of the suspensions of free products, as well as the conjugacy problem of certain automorphisms of free products.To be more precise, given a free product G=G1astdotsast Gpast Fk an automorphism phi is said atoroidal if no power fixes the conjugacy class of an hyperbolic element. It is called fully irreducible if the given free factor system [G1],dots,[Gp] is the largest one that is fixed by every power of the automorphism. It is said toral if for all i, there exists giin G such that {rm ad}{gi}circ phi|{Gi} is the identity on the free factor Gi. It is said to have central condition if for each i, there exists giin G conjugating phi(Gi) to Gi, and if there exists a non-trivial element of Girtimes{{rm ad}{gi} circ phi|{Gi}} mathbb{Z} that is central in Girtimes{{rm ad}{gi} circ phi|{Gi}} mathbb{Z}.We prove, in Theorem 4.28, that if phi is atoroidal and fully irreducible, and if the free product is non-elementary (kgeq 2 or p+k geq 3), the group Grtimesphi mathbb{Z} is relatively hyperbolic (relative to the mapping torus of each Gi). Then in Theorem 6.10 we prove the same result holds if phi is atoroidal with central condition. We also prove in Theorem 7.21 that if all Gi are abelian, the conjugacy problem is solvable for toral atoroidal automorphisms. These are analogue of the result of Brinkmann [7] (which gave the hyperbolicity result for free groups) and the result of Dahmani [12] (which solved the conjugacy problem of hyperbolic automorphisms).
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Ruoyu Li. Relative hyperbolicity of suspensions of free products. Commutative Algebra [math.AC]. Université Grenoble Alpes, 2018. English. ⟨NNT : 2018GREAM048⟩. ⟨tel-02010283⟩

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