On sequences of finitely generated discrete groups

Kapovich, Michael

doi:10.1090/conm/510/10024

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…In general, however, G is only finitely presented relative to P, and the action on T only satisfies a weaker property than stability, which we call hypostability (see [Kap10] for a different property called semistability).…”

Section: Hypostabilitymentioning

confidence: 99%

Splittings and automorphisms of relatively hyperbolic groups

Guirardel¹,

Levitt²

2015

Groups Geom. Dyn.

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We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL n (Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL n (Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups.Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P G) is infinite, then P is a vertex group in a splitting of G. If P is torsion-free, then Out(P G) is of type VF, in particular finitely presented.We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitely-ended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.

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Section: Hypostabilitymentioning

confidence: 99%

Splittings and automorphisms of relatively hyperbolic groups

Guirardel¹,

Levitt²

2015

Groups Geom. Dyn.

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“…It was suggested ( [20]) that these might be the only examples. Soon it was noticed ( [12]) that the boundary of the 3-dimensional Euclidean tetrahedron whose opposite edges are equal (or equivalently, whose four faces are congruent triangles) is also a 2-box. Later, it was conjectured ( [22]) that all n-boxes have to be isometric to a quotient of a flat torus by an action of a group of isometries which is isomorphic to a product of Z 2 -groups.…”

Section: Resultsmentioning

confidence: 99%

Number of extremal subsets in Alexandrov spaces and rigidity

Lebedeva¹

2014

ERA-MS

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In this paper we announce the following result. We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of R n by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.

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“…The proof of (1). The nonelementariness of follows from [21,Theorem 1.4]. Now, we come to prove that if is nondiscrete, then there is an element ∈ ( ) such that the subgroup ⟨ ⟩ is nondiscrete.…”

Section: Algebraic Convergencementioning

confidence: 99%

Discreteness and Convergence of Complex Hyperbolic Isometry Groups

2013

Abstract and Applied Analysis

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On sequences of finitely generated discrete groups

Cited by 7 publications

References 21 publications

Splittings and automorphisms of relatively hyperbolic groups

Splittings and automorphisms of relatively hyperbolic groups

Number of extremal subsets in Alexandrov spaces and rigidity

Discreteness and Convergence of Complex Hyperbolic Isometry Groups

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