685 research outputs found
Geometric Intersection Number and analogues of the Curve Complex for free groups
For the free group of finite rank we construct a canonical
Bonahon-type continuous and -invariant \emph{geometric intersection
form}
Here is the closure of unprojectivized Culler-Vogtmann's
Outer space in the equivariant Gromov-Hausdorff convergence topology
(or, equivalently, in the length function topology). It is known that
consists of all \emph{very small} minimal isometric actions of
on -trees. The projectivization of provides a
free group analogue of Thurston's compactification of the Teichm\"uller space.
As an application, using the \emph{intersection graph} determined by the
intersection form, we show that several natural analogues of the curve complex
in the free group context have infinite diameter.Comment: Revised version, to appear in Geometry & Topolog
Kleinian groups and the rank problem
We prove that the rank problem is decidable in the class of torsion-free
word-hyperbolic Kleinian groups. We also show that every group in this class
has only finitely many Nielsen equivalence classes of generating sets of a
given cardinality.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper12.abs.htm
Triangle inequalities in path metric spaces
We study side-lengths of triangles in path metric spaces. We prove that
unless such a space X is bounded, or quasi-isometric to line or half-line,
every triple of real numbers satisfying the strict triangle inequalities, is
realized by the side-lengths of a triangle in X. We construct an example of a
complete path metric space quasi-isometric to the Euclidean plane, for which
every degenerate triangle has one side which is shorter than a certain uniform
constant.Comment: 21 pages, 6 figure
Krull dimensions of rings of holomorphic functions
We prove that the Krull dimension of the ring of holomorphic functions of a
connected complex manifold is at least continuum if it is positive.Comment: 6 pages. An error pointed out by Pete Clark is corrected. The
stronger statement about the Krull dimension at least continuum is prove
Homological dimension and critical exponent of Kleinian groups
We prove that the relative homological dimension of a Kleinian group G does
not exceed 1 + the critical exponent of G. As an application of this result we
show that for a geometrically finite Kleinian group G, if the topological
dimension of the limit set of G equals its Hausdorff dimension, then the limit
set is a round sphere.Comment: 38 page
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