10 research outputs found
A Combination Theorem for Convex Hyperbolic Manifolds, with Applications to Surfaces in 3-Manifolds
We prove the convex combination theorem for hyperbolic n-manifolds.
Applications are given both in high dimensions and in 3 dimensions. One
consequence is that given two geometrically finite subgroups of a discrete
group of isometries of hyperbolic n-space, satisfying a natural condition on
their parabolic subgroups, there are finite index subgroups which generate a
subgroup that is an amalgamated free product. Constructions of infinite volume
hyperbolic n-manifolds are described by gluing lower dimensional manifolds. It
is shown that every slope on a cusp of a hyperbolic 3-manifold is a multiple
immersed boundary slope. If a 3-manifold contains a maximal surface group not
carried by an embedded surface then it contains the fundamental group of a book
of I-bundles with more than two pages.Comment: 43 pages, 10 Postscript figures. Minor changes to 2.4, 2.5, 8.1, 8.5.
Citations added and correcte
The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups
We show that if is a quasiconvex subgroup of infinite index in a
non-elementary hyperbolic group then the Schreier coset graph for
relative to is non-amenable (that is, has positive Cheeger constant).
We present some corollaries regading the Martin boundary and Martin
compactification of and the co-growth of in .Comment: updated versio
Combination of convergence groups
We state and prove a combination theorem for relatively hyperbolic groups
seen as geometrically finite convergence groups. For that, we explain how to
contruct a boundary for a group that is an acylindrical amalgamation of
relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our
result to Sela's theory on limit groups and prove their relative hyperbolicity.
We also get a proof of the Howson property for limit groups.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.htm
Doubles of groups and hyperbolic LERF 3-manifolds
We show that the quasiconvex subgroups in doubles of certain negatively
curved groups are closed in the profinite topology. This allows us to construct
the first known large family of hyperbolic 3-manifolds such that any finitely
generated subgroup of the fundamental group of any member of the family is
closed in the profinite topology.Comment: 32 pages, published versio
Asymptotic invariants, complexity of groups and related problems
We survey results about computational complexity of the word problem in
groups, Dehn functions of groups and related problems.Comment: 86 pages. Preliminary version, comments are welcome. v2: some
references added, misprints fixed, some changes suggested by the readers are
made. 88 pages. v3: more readers' suggestions implemented, index added, the
list of references improved. This version is submitted to a journal. v4: The
paper is accepted in Bulletin of Mathematical Science