138 research outputs found
Obstructions to nonpositive curvature for open manifolds
We study algebraic conditions on a group G under which every properly
discontinuous, isometric G-action on a Hadamard manifold has a G-invariant
Busemann function. For such G we prove the following structure theorem: every
open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy
equivalent to a finite complex of codimension >2 is an open regular
neighborhood of a subcomplex of the same codimension. In this setting we show
that each tangential homotopy type contains infinitely many open K(G,1)
manifolds that admit no complete nonpositively curved metric even though their
universal cover is the Euclidean space. A sample application is that an open
contractible manifold W is homeomorphic to a Euclidean space if and only if the
product of W and a circle admits a complete Riemannian metric of nonpositive
curvature.Comment: 29 page
Quasi-hyperbolic planes in relatively hyperbolic groups
We show that any group that is hyperbolic relative to virtually nilpotent
subgroups, and does not admit peripheral splittings, contains a
quasi-isometrically embedded copy of the hyperbolic plane. In natural
situations, the specific embeddings we find remain quasi-isometric embeddings
when composed with the inclusion map from the Cayley graph to the coned-off
graph, as well as when composed with the quotient map to "almost every"
peripheral (Dehn) filling.
We apply our theorem to study the same question for fundamental groups of
3-manifolds.
The key idea is to study quantitative geometric properties of the boundaries
of relatively hyperbolic groups, such as linear connectedness. In particular,
we prove a new existence result for quasi-arcs that avoid obstacles.Comment: v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4
figures. An application (Theorem 1.2) is weakened as there was an error in
its proof in section 7, all other changes minor, improved expositio
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface
(in particular, for any simple polygon with geodesic metric)
and any positive number , the minimum number of closed balls of radius
with centers at and covering the set is at most 19
times the maximum number of disjoint closed balls of radius centered
at points of : , where and
are the covering and the packing numbers of by -balls.Comment: 27 page
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