Combination of convergence groups

Dahmani, François

doi:10.2140/gt.2003.7.933

Cited by 189 publications

(286 citation statements)

References 29 publications

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“…Finally let s be an element of S . We take an arbitrary preimage g 2 G.0/ of s. Then the image of the element g becomes 7 parabolic at a certain step according to (v). Thus s is conjugate to an element of R in S .…”

Section: Outline Of the Methodsmentioning

confidence: 99%

Small cancellations over relatively hyperbolic groups and embedding theorems

2010

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We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer to the wellknown question of the existence of a finitely generated group G other than ‫ޚ2=ޚ‬ such that all nontrivial elements of G are conjugate.

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Section: Outline Of the Methodsmentioning

confidence: 99%

Small cancellations over relatively hyperbolic groups and embedding theorems

2010

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“…By [3] 1 .R/ is hyperbolic relative to 1 .B k /'s, so a combination theorem for relatively hyperbolic groups due to Dahmani [9] implies that 1 .DR/ is hyperbolic relative to 1 .B k /'s. Hence, by a recent result of Drutu-Sapir [17, Corollary 1.14] 1 .DR/ is hyperbolic relative to the images of…”

Section: Relative Strict Hyperbolizationmentioning

confidence: 99%

Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Belegradek

2006

Algebr. Geom. Topol.

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This paper contains examples of closed aspherical manifolds obtained as a byproduct of recent work by the author [3] on the relative strict hyperbolization of polyhedra. The following is proved.(I) Any closed aspherical triangulated n -manifold M n with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+1)-manifold N n+1 with hyperbolic fundamental group.(II) If B 1 , . . . B m are closed aspherical triangulated n -manifolds, then there is a closed aspherical triangulated manifold N of dimension n+1 such that N has nonzero simplicial volume, N retracts to each B k , and π 1 (N) is hyperbolic relative to π 1 (B k ) 's.(III) Any finite aspherical simplicial complex is a retract of a closed aspherical triangulated manifold with positive simplicial volume and non-elementary relatively hyperbolic fundamental group. 20F65

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“…The splitting Λ v of G v = H is the HNN extension associated to the semidirect product. The group G = π 1 (Θ 0 ) is hyperbolic relative to the nilpotent group H by [4], and it may be checked that Θ 0 is its JSJ decomposition over abelian (or nilpotent) groups relative to H. Let Λ 0 be obtained by refining Θ 0 using Λ v in the obvious way. Collapsing e 1 , e 2 in Λ yields an HNN extension Γ 0 with edge group Z 2 = a, b .…”

Section: Figure 2: Infinitely Many Refinements Of a Graph Of Groupsmentioning

confidence: 99%

“…Assertion 3 applies to abelian splittings (i.e. splittings over abelian groups) of limit groups, since by [1,4] limit groups are toral relatively hyperbolic (i.e. torsion-free and hyperbolic relative to finitely generated abelian groups).…”

Section: Introductionmentioning

confidence: 99%

Vertex finiteness for splittings of relatively hyperbolic groups

Guirardel¹,

Levitt²

2016

Isr. J. Math.

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Consider a group G and a family A of subgroups of G. We say that vertex finiteness holds for splittings of G over A if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in A.We show vertex finiteness when G is a toral relatively hyperbolic group and A is the family of abelian subgroups.We also show vertex finiteness when G is hyperbolic relative to virtually polycyclic subgroups and A is the family of virtually cyclic subgroups; if moreover G is one-ended, there are only finitely many minimal G-trees with virtually cyclic edge stabilizers, up to automorphisms of G.

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Combination of convergence groups

Cited by 189 publications

References 29 publications

Small cancellations over relatively hyperbolic groups and embedding theorems

Small cancellations over relatively hyperbolic groups and embedding theorems

Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Vertex finiteness for splittings of relatively hyperbolic groups

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