Trees of cylinders and canonical splittings

Guirardel, Vincent; Levitt, Gilbert

doi:10.2140/gt.2011.15.977

Cited by 43 publications

(104 citation statements)

References 27 publications

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“…We do not have an exact sequence as in Theorem 1.4 because there is no Out(G; P)-invariant splitting. In order to prove Theorems 1.2 and 1.3, we use the tree of cylinders introduced in [GL11] to obtain a non-trivial splitting over finite groups which is invariant or has an infinite group of twists (Corollary 7.11).…”

Section: Moreover One May Decide Algorithmically Whether Out(g) Is Fmentioning

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: Moreover One May Decide Algorithmically Whether Out(g) Is Fmentioning

confidence: 99%

Splittings and automorphisms of relatively hyperbolic groups

Guirardel¹,

Levitt²

2015

Groups Geom. Dyn.

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“…If v is a vertex of T with G v non-abelian, it belongs to at least two cylinders, so G v is a vertex stabilizer of T c . The tree T c is 2-acylindrical (Proposition 6.3 of [14]), and one can make it reduced by collapsing edges (this does not change non-abelian vertex stabilizers).…”

Section: Splittings With Several Edgesmentioning

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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“…In [Guirardel and Levitt 2008], we associated a tree of cylinders T c to any Ᏹ-tree T , as follows. Two (nonoriented) edges of T are equivalent if their stabilizers are commensurable.…”

Section: Amentioning

confidence: 99%

“…We have shown a general construction producing a canonical splitting T c from a canonical deformation space: the tree of cylinders [Guirardel and Levitt 2008]. It also enjoys strong compatibility properties.…”

Section: Introductionmentioning

confidence: 99%