Results 1  10
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24
DEFORMATION SPACES OF TREES
, 2007
"... Let G be a finitely generated group. Two simplicial Gtrees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include CullerVogtmann’s outer space, and spaces of JSJ decompositions. We discuss ..."
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Cited by 21 (2 self)
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Let G be a finitely generated group. Two simplicial Gtrees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include CullerVogtmann’s outer space, and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.
Boundaries of hyperbolic groups
 CONTEMPORARY MATHEMATICS
"... In this paper we survey the known results about boundaries of wordhyperbolic groups. ..."
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Cited by 13 (0 self)
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In this paper we survey the known results about boundaries of wordhyperbolic groups.
Limits of (certain) CAT(0) groups, II: The Hopf property and the shortening argument
, 2004
"... This is the second in a series of papers about torsionfree groups which act properly and cocompactly on a CAT(0) metric space with isolated flats and relatively thin triangles. Our approach is to adapt the methods of Sela and others for wordhyperbolic groups to this context of nonpositive curvat ..."
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Cited by 6 (1 self)
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This is the second in a series of papers about torsionfree groups which act properly and cocompactly on a CAT(0) metric space with isolated flats and relatively thin triangles. Our approach is to adapt the methods of Sela and others for wordhyperbolic groups to this context of nonpositive curvature. The main result in this paper is that (under certain technical hypotheses) such a group as above is Hopfian. This (mostly) answers a question of Sela.
A generalization of the LyndonHochschildSerre spectral sequence with applications to group cohomology and decompositions of groups
 J. Group Theory
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Minimal cubings
, 2008
"... We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev’s original construction. We also show that almost invariant sets can be chosen to be ..."
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Cited by 3 (2 self)
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We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev’s original construction. We also show that almost invariant sets can be chosen to be in very good position. Let G be a finitely generated group, and let H1,...,Hn be subgroups. For i = 1,...,n, let Xi be a nontrivial Hi–almost invariant subset of G. In [6], Sageev gave a natural construction of a cubing C(X1,...,Xn) with a G–action which reflects the way in which the translates of the Xi’s intersect each other. In order to give the reader a feel for this, we start by discussing a simple and closely related topological example. For other simple examples, the reader ∗ Partially supported by NSF grants DMS 034681 and 9626537 1 is referred to Sageev’s paper [6]. Consider a finite family F = {S1,...,Sn} of compact curves in general position on an orientable surface M. There is a
RANK GRADIENT OF CYCLIC COVERS
"... Abstract. If M is an orientable hyperbolic 3manifold with finite volume and φ: π1(M) ↠ Z, the family of covers corresponding to {φ−1(nZ)  n ∈ N} has rank gradient 0 if and only if the Poincaré–Lefschetz dual of the class in H1 (M; Z) corresponding to φ is represented by a fiber. This generalizes ..."
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Abstract. If M is an orientable hyperbolic 3manifold with finite volume and φ: π1(M) ↠ Z, the family of covers corresponding to {φ−1(nZ)  n ∈ N} has rank gradient 0 if and only if the Poincaré–Lefschetz dual of the class in H1 (M; Z) corresponding to φ is represented by a fiber. This generalizes a theorem of M. Lackenby. If M is closed, we give an explicit lower bound on the rank gradient. The proof uses an acylindrical accessibility theorem due to R. Weidmann and the following result: if M is a closed, orientable hyperbolic 3manifold and S is a connected incompressible surface that is not a fiber or semifiber of M, then the π1Maction on the tree determined by S is (14g − 12)acylindrical, where g is the genus of S. By the rank of a manifold M, rk M, we will refer to the rank of its fundamental group; that is, the minimal cardinality of a generating set. Given a fixed closed manifold M, the rank gradient of a family of covers {Mn → M}, each with finite degree, is defined as rg {Mn}. = inf
Strong accessibility for hyperbolic groups
, 2007
"... We use an accessibility result of Delzant and Potyagailo to prove Swarup’s Strong Accessibility Conjecture for Gromov hyperbolic groups with no 2torsion. A particular notion of a hierarchy for 3manifolds motivated Swarup to make the following conjecture. Swarup’s Strong Accessibility Conjecture Le ..."
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Cited by 2 (0 self)
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We use an accessibility result of Delzant and Potyagailo to prove Swarup’s Strong Accessibility Conjecture for Gromov hyperbolic groups with no 2torsion. A particular notion of a hierarchy for 3manifolds motivated Swarup to make the following conjecture. Swarup’s Strong Accessibility Conjecture Let G be a hyperbolic group. Decompose G maximally over finite subgroups, and then take the resulting vertex groups, and decompose those maximally over twoended subgroups. Now repeat this process on the new vertex groups and so on. Then this process must eventually terminate, with subgroups of G that are unsplittable over finite and twoended subgroups. We prove this result in the case that G has no 2torsion, making heavy use of the main result from [DP01]. In [DP01], Delzant and Potyagailo show the existence of finite hierarchies for finitely presented groups with no 2torsion, over
Some open 3manifolds and 3orbifolds without locally finite canonical decompositions
, 2008
"... We give examples of open 3manifolds and 3orbifolds that exhibit pathological behavior with respect to splitting along surfaces (2suborbifolds) with nonnegative Euler characteristic. ..."
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We give examples of open 3manifolds and 3orbifolds that exhibit pathological behavior with respect to splitting along surfaces (2suborbifolds) with nonnegative Euler characteristic.