64 research outputs found
Hyperbolic and cubical rigidities of Thompson's group V
In this article, we state and prove a general criterion allowing us to show
that some groups are hyperbolically elementary, meaning that every isometric
action of one of these groups on a Gromov-hyperbolic space either fixes a point
at infinity or has bounded orbits. Also, we show how such a hyperbolic rigidity
leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As
an application, we prove that Thompson's group is hyperbolically
elementary, and we deduce that it satisfies Property , ie.,
every isometric action of on a finite-dimensional CAT(0) cube complex fixes
a point. It provides the first example of a (finitely presented) group acting
properly on an infinite-dimensional CAT(0) cube complex such that all its
actions on finite-dimensional CAT(0) cube complexes have global fixed points.Comment: 24 pages, 5 figures. Comments are welcom
Coning-off CAT(0) cube complexes
In this paper, we study the geometry of cone-offs of CAT(0) cube complexes
over a family of combinatorially convex subcomplexes, with an emphasis on their
Gromov-hyperbolicity. A first application gives a direct cubical proof of the
characterization of the (strong) relative hyperbolicity of right-angled Coxeter
groups, which is a particular case of a result due to Behrstock, Caprace and
Hagen. A second application gives the acylindrical hyperbolicity of
small cancellation quotients of free products.Comment: 45 pages, 13 figures. Comments are welcom
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