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Results tagged with alexandrov-geometry
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user 1463
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 $\mathrm{CAT}[\kappa]$).
9
votes
Amalgamated product acting on CAT(0) cube complex
To extend the gluing result from Bridson--Haefliger to non-positively curved cube complexes, it is important to work in the correct category.
If we want the result to also be a non-positively curved c …
- 25.2k
3
votes
Gluing hexagons to get a locally CAT(0) space
The fourth example was studied by Brady and Crisp in their CMH paper CAT(0) and CAT(-1) dimensions of torsion-free hyperbolic groups, so it would be reasonable to call its fundamental group the "Brady …
- 25.2k
13
votes
Accepted
Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite
Here's a proof.
Lemma: Suppose $G$ is a (word-)hyperbolic group acting properly discontinously, cocompactly and faithfully on a CAT(0) space $X$. Then there is a uniform bound $R_0$ on the width …
- 25.2k
21
votes
When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?
An example of a free-by-cyclic group that is not CAT(0) was given by Gersten. It is constructed from the automorphism of $F_3\cong\langle a,b,c\rangle$ that sends
$a\mapsto a,~b\mapsto ba,~ c\mapsto …
- 25.2k
6
votes
Braid groups acting on CAT(0)-complexes
Here's question 2.16 of Bestvina's problem list:
Q 2.16. (Ruth Charney) Are all [finite type] Artin groups CAT(0)?
The answer is yes for small numbers of generators by the work of Krammer, Tom Brady …
- 25.2k
5
votes
Convex hull in CAT(0)
It follows quickly from the definition that closed balls are convex.
[Proof: Let p,q be in the ball of radius R about o, and let x lie on the geodesic from p to q. Then $d(o,x)\leq d(\bar{o},\bar{x} …
- 25.2k