7,759 research outputs found
Relative second bounded cohomology of free groups
This paper is devoted to the computation of the space
, where is a free group of finite rank
and is a subgroup of finite rank. More precisely we prove that
has infinite index in if and only if
is not trivial, and furthermore, if and only if there is an isometric embedding
, where is the space of
bounded alternating functions on equipped with the defect norm.Comment: 14 pages. small corrections; this version has been accepted for
publication by Geometriae Dedicat
The Patterson-Sullivan embedding and minimal volume entropy for outer space
Motivated by Bonahon's result for hyperbolic surfaces, we construct an
analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann
outer space into the space of projectivized geodesic currents on a
free group. We prove that this map is a topological embedding. We also prove
that for every the minimum of the volume entropy of the universal
covers of finite connected volume-one metric graphs with fundamental group of
rank and without degree-one vertices is equal to and that
this minimum is realized by trivalent graphs with all edges of equal lengths,
and only by such graphs.Comment: An updated versio
Deformation and rigidity of simplicial group actions on trees
We study a notion of deformation for simplicial trees with group actions
(G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a
deformation if there is a finite sequence of collapse and expansion moves
joining them. We show that this relation on the set of G-trees has several
characterizations, in terms of dynamics, coarse geometry, and length functions.
Next we study the deformation space of a fixed G-tree X. We show that if X is
`strongly slide-free' then it is the unique reduced tree in its deformation
space.
These methods allow us to extend the rigidity theorem of Bass and Lubotzky to
trees that are not locally finite. This yields a unique factorization theorem
for certain graphs of groups. We apply the theory to generalized
Baumslag-Solitar groups and show that many have canonical decompositions. We
also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper8.abs.htm
Ending Laminations and Cannon-Thurston Maps
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian
surface groups. In this paper we prove that pre-images of points are precisely
end-points of leaves of the ending lamination whenever the Cannon-Thurston map
is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one.
This completes the proof of the conjectural picture of Cannon-Thurston maps for
surface groups.Comment: v4: Final version 22pgs 2figures. Includes the main theorem of the
appendix arXiv:1002.2090 by Shubhabrata Das and Mahan Mj. To appear in
Geometric and Functional Analysi
Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes
We introduce coordinates on the moduli spaces of maximal globally hyperbolic
constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are
derived from the parametrisation of the moduli spaces by the bundle of measured
geodesic laminations over Teichm\"uller space of S and can be viewed as
analytic continuations of the shear coordinates on Teichm\"uller space. In
terms of these coordinates the gravitational symplectic structure takes a
particularly simple form, which resembles the Weil-Petersson symplectic
structure in shear coordinates, and is closely related to the cotangent bundle
of Teichm\"uller space. We then consider the mapping class group action on the
moduli spaces and show that it preserves the gravitational symplectic
structure. This defines three distinct mapping class group actions on the
cotangent bundle of Teichm\"uller space, corresponding to different values of
the curvature.Comment: 40 pages, 6 figure
Convex cocompact subgroups of mapping class groups
We develop a theory of convex cocompact subgroups of the mapping class group
MCG of a closed, oriented surface S of genus at least 2, in terms of the action
on Teichmuller space. Given a subgroup G of MCG defining an extension L_G:
1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic
group then G is a convex cocompact subgroup of MCG. When G is free and convex
cocompact, called a "Schottky subgroup" of MCG, the converse is true as well; a
semidirect product of pi_1(S) by a free group G is therefore word hyperbolic if
and only if G is a Schottky subgroup of MCG. The special case when G=Z follows
from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance:
sufficiently high powers of any independent set of pseudo-Anosov mapping
classes freely generate a Schottky subgroup.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper5.abs.htm
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