20 research outputs found
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
No-splitting property and boundaries of random groups
We prove that random groups in the Gromov density model, at any density,
satisfy property (FA), i.e. they do not act non-trivially on trees. This
implies that their Gromov boundaries, defined at density less than 1/2, are
Menger curves.Comment: 20 page
Peripheral fillings of relatively hyperbolic groups
A group theoretic version of Dehn surgery is studied. Starting with an
arbitrary relatively hyperbolic group we define a peripheral filling
procedure, which produces quotients of by imitating the effect of the Dehn
filling of a complete finite volume hyperbolic 3--manifold on the
fundamental group . The main result of the paper is an algebraic
counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that
peripheral subgroups of 'almost' have the Congruence Extension Property and
the group is approximated (in an algebraic sense) by its quotients obtained
by peripheral fillings. Various applications of these results are discussed.Comment: The difference with the previous version is that Proposition 3.2 is
proved for quasi--geodesics instead of geodesics. This allows to simplify the
exposition in the last section. To appear in Invent. Mat
Subset currents on free groups
We introduce and study the space of \emph{subset currents} on the free group
. A subset current on is a positive -invariant locally finite
Borel measure on the space of all closed subsets of consisting of at least two points. While ordinary geodesic currents
generalize conjugacy classes of nontrivial group elements, a subset current is
a measure-theoretic generalization of the conjugacy class of a nontrivial
finitely generated subgroup in , and, more generally, in a word-hyperbolic
group. The concept of a subset current is related to the notion of an
"invariant random subgroup" with respect to some conjugacy-invariant
probability measure on the space of closed subgroups of a topological group. If
we fix a free basis of , a subset current may also be viewed as an
-invariant measure on a "branching" analog of the geodesic flow space for
, whose elements are infinite subtrees (rather than just geodesic lines)
of the Cayley graph of with respect to .Comment: updated version; to appear in Geometriae Dedicat
Intersection form, laminations and currents on free groups
Let be a free group of rank , let be a geodesic current
on and let be an -tree with a very small isometric action
of . We prove that the geometric intersection number is equal
to zero if and only if the support of is contained in the dual algebraic
lamination of . Applying this result, we obtain a generalization of
a theorem of Francaviglia regarding length spectrum compactness for currents
with full support. As another application, we define the notion of a
\emph{filling} element in and prove that filling elements are "nearly
generic" in . We also apply our results to the notion of \emph{bounded
translation equivalence} in free groups.Comment: revised version, to appear in GAF
Bowditch's JSJ tree and the quasiâisometry classification of certain Coxeter groups
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry
invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our
main result gives an explicit, computable "visual" construction of this tree
for certain hyperbolic right-angled Coxeter groups. As an application of our
construction we identify a large class of such groups for which the JSJ tree,
and hence the visual boundary, is a complete quasi-isometry invariant, and thus
the quasi-isometry problem is decidable. We also give a direct proof of the
fact that among the Coxeter groups we consider, the cocompact Fuchsian groups
form a rigid quasi-isometry class. In an appendix, written jointly with
Christopher Cashen, we show that the JSJ tree is not a complete quasi-isometry
invariant for the entire class of Coxeter groups we consider.Comment: 46 pages, 5 figures. Added Appendix B (joint with Christopher Cashen)
and a discussion about generalizing the results in the paper to higher
dimensions. Made other minor revisions based on the referee's comments. To
appear in Journal of Topolog