103 research outputs found
From wall spaces to CAT(0) cube complexes
We explain how to adapt a construction due to M. Sageev in order to construct a proper action of a group on a CAT(0) cube complex starting from a proper action of the group on a wall space
Boundary curves of surfaces with the 4-plane property
Let M be an orientable and irreducible 3-manifold whose boundary is an
incompressible torus. Suppose that M does not contain any closed nonperipheral
embedded incompressible surfaces. We will show in this paper that the immersed
surfaces in M with the 4-plane property can realize only finitely many boundary
slopes. Moreover, we will show that only finitely many Dehn fillings of M can
yield 3-manifolds with nonpositive cubings. This gives the first examples of
hyperbolic 3-manifolds that cannot admit any nonpositive cubings.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper21.abs.htm
Line Patterns in Free Groups
We study line patterns in a free group by considering the topology of the
decomposition space, a quotient of the boundary at infinity of the free group
related to the line pattern. We show that the group of quasi-isometries
preserving a line pattern in a free group acts by isometries on a related space
if and only if there are no cut pairs in the decomposition space.Comment: 35 pages, 22 figures, PDFLatex; v2. finite index requires extra
hypothesis; v3. 37 pages, 24 figures: updated references and add example in
Section 6.3 of a rigid pattern for which the free group is not finite index
in the group of pattern preserving quasi-isometries; v4. 40 pages, 26
figures: improved exposition and add example in Section 6.4 of a rigid
pattern whose cube complex is not a tre
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