1,101 research outputs found
Actions of certain arithmetic groups on Gromov hyperbolic spaces
We study the variety of actions of a fixed (Chevalley) group on arbitrary
geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete
classification. In rank one, we obtain some partial results and give a
conjectural picture.Comment: v1: 23 pages, 4 figures. v2: 24 pages, 4 figures. Fixed a bad typo in
Claim 3.3 and made some other small changes. v3: A few small clarifications.
To appear in AG
Deciding Isomorphy using Dehn fillings, the splitting case
We solve Dehn's isomorphism problem for virtually torsion-free relatively
hyperbolic groups with nilpotent parabolic subgroups.
We do so by reducing the isomorphism problem to three algorithmic problems in
the parabolic subgroups, namely the isomorphism problem, separation of torsion
(in their outer automorphism groups) by congruences, and the mixed Whitehead
problem, an automorphism group orbit problem. The first step of the reduction
is to compute canonical JSJ decompositions. Dehn fillings and the given
solutions of the algorithmic problems in the parabolic groups are then used to
decide if the graphs of groups have isomorphic vertex groups and, if so,
whether a global isomorphism can be assembled.
For the class of finitely generated nilpotent groups, we give solutions to
these algorithmic problems by using the arithmetic nature of these groups and
of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections.
The main changes to the previous version are a better treatment of the
algorithmic recognition and presentation of virtually cyclic subgroups and a
new proof of a rigidity criterion obtained by passing to a torsion-free
finite index subgroup. The previous proof relied on an incorrect result. To
appear in Inventiones Mathematica
Infinite groups with fixed point properties
We construct finitely generated groups with strong fixed point properties.
Let be the class of Hausdorff spaces of finite covering
dimension which are mod- acyclic for at least one prime . We produce the
first examples of infinite finitely generated groups with the property that
for any action of on any , there is a global fixed
point. Moreover, may be chosen to be simple and to have Kazhdan's property
(T). We construct a finitely presented infinite group that admits no
non-trivial action by diffeomorphisms on any smooth manifold in
. In building , we exhibit new families of hyperbolic
groups: for each and each prime , we construct a non-elementary
hyperbolic group which has a generating set of size , any proper
subset of which generates a finite -group.Comment: Version 2: 29 pages. This is the final published version of the
articl
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
We introduce and study the notions of hyperbolically embedded and very
rotating families of subgroups. The former notion can be thought of as a
generalization of the peripheral structure of a relatively hyperbolic group,
while the later one provides a natural framework for developing a geometric
version of small cancellation theory. Examples of such families naturally occur
in groups acting on hyperbolic spaces including hyperbolic and relatively
hyperbolic groups, mapping class groups, , and the Cremona group.
Other examples can be found among groups acting geometrically on
spaces, fundamental groups of graphs of groups, etc. We obtain a number of
general results about rotating families and hyperbolically embedded subgroups;
although our technique applies to a wide class of groups, it is capable of
producing new results even for well-studied particular classes. For instance,
we solve two open problems about mapping class groups, and obtain some results
which are new even for relatively hyperbolic groups.Comment: Revision, corrections and improvement of the expositio
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