Asymmetry of Outer Space of a free product

Syrigos, Dionysios

doi:10.48550/arxiv.1507.01849

Cited by 2 publications

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“…Recently, there are more papers that they indicate that we can find more similarities between CV n and O. For example, the construction of hyperbolic spaces on which Out(G, {H 1 , ..., H r }) acts ( [10]), [13]), the boundary of outer space ( [12]), the Tits alternative for subgroups of Out(G) ( [14]), the study of the asymmetry of the Lipschitz metric( [20]) and the study of the centralisers of IWIP automorphisms( [19]). OUTLINE: In Section 2, we recall some useful definitions and facts, in additional we generalise some well known notions for free groups to the free product case and we prove some basic preliminary results that we need for the main theorem.…”

Section: Introductionmentioning

confidence: 99%

Stabiliser of an Attractive Fixed Point of an IWIP Automorphism of a free product

Syrigos

2016

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For a group G of finite Kurosh rank and for some arbiratily free product decomposition of G, G = H 1 * H 2 * ... * H r * F q , where F q is a finitely generated free group, we can associate some (relative) outer space O(G, {H 1 , ..., H r }). We define the relative boundary ∂(G, {H 1 , ..., H r }) = ∂(G, O) corresponding to the free product decomposition, as the set of infinite reduced words (with respect to free product length). By denoting Out(G, {H 1 , ..., H r }) the subgroup of Out(G) which is consisted of the outer automorphisms which preserve the set of conjugacy classes of H i 's, we prove that for the stabiliser Stab(X) of an attractive fixed point in X ∈ ∂(G, {H 1 , ..., H r }) of an irreducible with irreducible powers automorphism relative to O, it holds that it has a (normal) subgroup B isomorphic to subgroup of r i=1Out(H i ) such that Stab(X)/B is isomorphic to Z. The proof relies heavily on the machinery of the attractive lamination of an IWIP automorphism relative to O.

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Section: Introductionmentioning

confidence: 99%

Stabiliser of an Attractive Fixed Point of an IWIP Automorphism of a free product

Syrigos

2016

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“…In addition, further motivation is that we can find natural generalisations for a lot of facts about CV n in the general case, for example in [10], Francaviglia and Martino generalised a lot of tools like train track maps and the Lipschitz metric. But there are some recent papers that show we can also use further methods of studying Out(F n) for Out(G) (where G is written as free product as above) such that the closure of outer space, the Tits alternatives for Out(G), the hyperbolic complex corresponding to Out(G) and the asymmetry of the outer space of a free product (see [14], [15], [16], [19] and [24]). Finally, the author as an application of the results of the present paper generalises a result of Hilion ([13]), about the stabiliser of attractive fixed point of an IWIP automorphism ( [25]).…”

Section: Introductionmentioning

confidence: 99%

Irreducible laminations for IWIP Automorphisms of free products and Centralisers

Syrigos¹

2014

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For every free product decomposition G = G 1 * ... * G q * F r , where F r is a finitely generated free group, of a group G of finite Kurosh rank, we can associate some (relative) outer space O. In this paper, we develop the theory of (stable) laminations for (relative) irreducible with irreducible powers (IWIP) automorphisms. In particular, we examine the action of Out(G, O) ≤ Out(G) (i.e. the automorphisms which preserve the set of conjugacy classes of G i 's) on the set of laminations. We generalise the theory of the attractive laminations associated to automorphisms of finitely generated free groups. The strategy is the same as in the classical case (see [1]), but some statements are slightly different because of the existence of the G i 's. More precisely, we prove that the stabiliser of the lamination of a relative IWIP is a Z-extension of a subgroup that is consisted of virtually elliptic automorphisms. Note that in the free case, virtually elliptic automorphisms are exactly the finite order automorphisms of Out(F n ). As a corollary of the previous theorem, we generalise the fact that the centraliser of an IWIP automorphism of a free group, is virtually cyclic. As a direct corollary, if Out(G) is virtually torsion free and every Aut(G i ) is finite, we prove that the centraliser of an IWIP is virtually cyclic. Finally, we give an example which shows that we cannot expect that in general the centraliser of a relative IWIP (and as a consequence the stabiliser of its stable lamination) is virtually cyclic.

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Asymmetry of Outer Space of a free product

Cited by 2 publications

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Stabiliser of an Attractive Fixed Point of an IWIP Automorphism of a free product

Stabiliser of an Attractive Fixed Point of an IWIP Automorphism of a free product

Irreducible laminations for IWIP Automorphisms of free products and Centralisers

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