Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Groves, Daniel

doi:10.2140/gt.2005.9.2319

Cited by 57 publications

(85 citation statements)

References 28 publications

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“…The exception to this is the proof that the groups under consideration are Hopfian. This result follows immediately from the author's paper [23,Theorem 5.2] (and the proof there does not depend on anything left out of this version of this paper). Since the proof of the Hopf property is technical, and no more enlightening than the proof of [23,Theorem 5.2], we chose to leave this result out of this paper.…”

Section: F65; 20f67 20e08 57m07mentioning

confidence: 60%

“…This result follows immediately from the author's paper [23,Theorem 5.2] (and the proof there does not depend on anything left out of this version of this paper). Since the proof of the Hopf property is technical, and no more enlightening than the proof of [23,Theorem 5.2], we chose to leave this result out of this paper. Leaving this proof out made it natural to take the results of [22] and the first version of this paper and merge them into this single paper.…”

Section: F65; 20f67 20e08 57m07mentioning

confidence: 60%

“…In a continuation paper [23] we use the results of this paper and of [48] to construct Makanin-Razborov diagrams for such a group . It is our hope that much, possibly all, of Sela's program can be carried out for these groups.…”

Section: F65; 20f67 20e08 57m07mentioning

confidence: 99%

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Limit groups for relatively hyperbolic groups. I. The basic tools

Groves

2009

Algebr. Geom. Topol.

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We begin the investigation of Γ-limit groups, where Γ is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of [16], we adapt the results from [22]. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate homomorphisms {h n : G → Γ}, we extract an R-tree with a nontrivial isometric G-action.We then provide an analogue of Sela's shortening argument.

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Section: F65; 20f67 20e08 57m07mentioning

confidence: 60%

Section: F65; 20f67 20e08 57m07mentioning

confidence: 60%

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Limit groups for relatively hyperbolic groups. I. The basic tools

Groves

2009

Algebr. Geom. Topol.

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“…The second problem is that the proof of Lemma 3.6 does not apply in this context. Since this paper was written, Groves has solved this problem [11].…”

Section: Proof Of the Main Theorem If ƒ 2 Limmentioning

confidence: 99%

Makanin–Razborov diagrams for limit groups

Alibegović

2007

Geom. Topol.

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“…As R i 's are all torsion-free hyperbolic groups relative to abelian subgroups (see the reference and the explanation in the proof of Lemma 4.1), and the structure of Hom.G; R i / can be described using the Makanin-Razborov diagram by the work of Groves [7] (cf Sela [14] for an original version for free groups). In this sense, Theorem 1.1 implies a uniform description of the structure of homomorphisms from a finitely generated group to torsion-free Kleinian groups of uniformly bounded volume.…”

Section: Introductionmentioning

confidence: 99%

A Jørgensen–Thurston theorem for homomorphisms

Liu

2012

Algebr. Geom. Topol.

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In this note, we provide a description of the structure homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jørgensen-Thurston Theorem in hyperbolic geometry. YI LIUfinitely-generated case can be reduced to the finitely-presented case, (Proposition 4.12).Acknowledgement. The author thanks Ian Agol and Daniel Groves for helpful conversations. PreliminariesIn this section, we recall some notions and results related to the topic of this paper.2.1. The Jørgensen-Thurston Theorem. For convenience of our discussion, we adopt the following notations and state the results in terms of Kleinian groups. Cf. [Th] for facts mentioned in this subsection.Let Γ be a torsion-free Kleinian group of finite covolume, namely, which has a fundamental domain in H 3 of finite volume. Then Γ has at most finitely many of conjugacy-classes parabolic subgroups, represented by subgroups P 1 , • • • , P q , (q ≥ 0), each isomorphic to a rank-2 free abelian group. We often call these chosen subgroups cusp representatives of Γ. By a slope-tupleor ambiguously, w.r.t. Γ), we shall mean that for each 1 ≤ j ≤ q, the j-th component ζ j is either trivial or a primitive element in P j . For any slopetuple ζ of Γ, we denote the Dehn filling of Γ along ζ as:i.e. Γ quotienting out the normal closure of the ζ j 's, and often denote the quotient epimorphism as:By Thurston's Hyperbolic Dehn-Filling Theorem ([Th, Theorem 5.8.2]), for generic slope-tuples ζ (avoiding finitely many primitive choices for each component), Γ ζ is isomorphic to a torision-free Kleinian group of finite covolume, indeed, strictly less than that of Γ if ζ is nonempty. These Γ ζ 's are usually called hyperbolic Dehn fillings of Γ. In fact, one may choose faithful Kleinian representations ρ ζ : Γ ζ → PSL 2 (C), so that for any sequence {ζ n } of slope-tuples, there is a subsequence for which the induced representations {ρ ζn • ι ζn } of Γ strongly converges. Moreover, if for each j-th component, {ζ j n } has no bounded subsequence of primitive elements in P j , then {ρ ζn • ι ζn } strongly converges to the inclusion Γ ⊂ PSL 2 (C).The following rephrases [Th, Theorem 5.12.1]:Theorem 2.1 (Jørgensen-Thurston). For any V > 0, there exist finitely many torsion-free Kleinian groups Γ 1 , • • • , Γ k of covolume at most V , (together with chosen cusp representatives), such that any torsion-free Kleinian group of covolume at most V is isomorphic to the hyperbolic Dehn filling (Γ i ) ζ of some Γ i along some slope-tuple ζ of Γ i .Remark 2.2. It is also implied from the proof that hyperbolic Dehn fillings decreases the covolume.

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Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Cited by 57 publications

References 28 publications

Limit groups for relatively hyperbolic groups. I. The basic tools

Limit groups for relatively hyperbolic groups. I. The basic tools

Makanin–Razborov diagrams for limit groups

A Jørgensen–Thurston theorem for homomorphisms

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