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Results tagged with alexandrov-geometry
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Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).
18
votes
Accepted
Mapping class group and CAT(0) spaces
(1) Bridson showed that if a mapping class group of a surface (of genus at least 3) acts on a CAT(0) space, then Dehn twists act as elliptic or parabolic elements. This implies that the mapping class …
- 68.9k
17
votes
Accepted
Gluing hexagons to get a locally CAT(0) space
The third example is a spine of the Gieseking manifold. The existence of a spine of this sort follows from a result of Iain Aitchison and the fact that the Gieseking is make of a single regular ideal …
- 68.9k
17
votes
Accepted
Metric spheres in CAT(0) manifolds
The answer is no. By the double suspension theorem of Cannon and Edwards, if $X^n$ is a homology $n$-sphere, then the double suspension $S^2X$ is a sphere. In particular, the cone $CSX$ on $SX$ will …
- 68.9k
13
votes
When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?
If you take the mapping torus of an automorphism of a surface with boundary, then it has a non-positively curved metric by a result of Leeb. Such automorphisms though will be sparse in the set of all …
- 68.9k