75 research outputs found
Core and intersection number for group actions on trees
We present the construction of some kind of "convex core" for the product of
two actions of a group on \bbR-trees. This geometric construction allows to
generalize and unify the intersection number of two curves or of two measured
foliations on a surface, Scott's intersection number of splittings, and the
apparition of surfaces in Fujiwara-Papasoglu's construction of the JSJ
splitting. In particular, this construction allows a topological interpretation
of the intersection number analogous to the definition for curves in surfaces.
As an application of this construction, we prove that an irreducible
automorphism of the free group whose stable and unstable trees are geometric,
is actually induced a pseudo-Anosov homeomorphism on a surface
The outer space of a free product
We associate a contractible ``outer space'' to any free product of groups
G=G_1*...*G_q. It equals Culler-Vogtmann space when G is free,
McCullough-Miller space when no G_i is Z. Our proof of contractibility (given
when G is not free) is based on Skora's idea of deforming morphisms between
trees.
Using the action of Out(G) on this space, we show that Out(G) has finite
virtual cohomological dimension, or is VFL (it has a finite index subgroup with
a finite classifying space), if the groups G_i and Out(G_i) have similar
properties. We deduce that Out(G) is VFL if G is a torsion-free hyperbolic
group, or a limit group (finitely generated fully residually free group).Comment: Updated reference. To appear in Proc. L.M.
A very short proof of Forester's rigidity result
The deformation space of a simplicial G-tree T is the set of G-trees which
can be obtained from T by some collapse and expansion moves, or equivalently,
which have the same elliptic subgroups as T. We give a short proof of a
rigidity result by Forester which gives a sufficient condition for a
deformation space to contain an Aut(G)-invariant G-tree. This gives a
sufficient condition for a JSJ splitting to be invariant under automorphisms of
G. More precisely, the theorem claims that a deformation space contains at most
one strongly slide-free G-tree, where strongly slide-free means the following:
whenever two edges e_1, e_2 incident on a same vertex v are such that G_{e_1}
is a subset of G_{e_2}, then e_1 and e_2 are in the same orbit under G_v.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper10.abs.htm
The isomorphism problem for all hyperbolic groups
We give a solution to Dehn's isomorphism problem for the class of all
hyperbolic groups, possibly with torsion. We also prove a relative version for
groups with peripheral structures. As a corollary, we give a uniform solution
to Whitehead's problem asking whether two tuples of elements of a hyperbolic
group are in the same orbit under the action of \Aut(G). We also get an
algorithm computing a generating set of the group of automorphisms of a
hyperbolic group preserving a peripheral structure.Comment: 71 pages, 4 figure
Deformation spaces of trees
Let G be a finitely generated group. Two simplicial G-trees 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
Culler-Vogtmann'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.Comment: Update to published version. 43 page
Splittings and automorphisms of relatively hyperbolic groups
We study automorphisms of a relatively hyperbolic group G. When G is
one-ended, 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 to G) of automorphisms of P that extend to
G: it is virtually a McCool group. If Out(P to G) is infinite, then P is a
vertex group in a splitting of G. If P is torsion-free, then Out(P to 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.Comment: Minor modifications. To appear in Geometry Groups and Dynamic
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