6
votes
1answer
134 views

Asymmetric metrics and cohomology

If $(X,d)$ is a metric space and $f : X \rightarrow \mathbb{R}$ is a Lipschitz function with Lipschitz constant $k < 1$, then the function $$ D(x,y) := d(x,y) + f(y) - f(x) $$ defines an asymmetric ...
10
votes
1answer
227 views

Mapping class group and CAT(0) spaces

I hope the questions are not too vague. Is the mapping class group of an orientable punctured surface $CAT(0)$ ? Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
7
votes
0answers
366 views

Uniquely geodesic groups

Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space. Examples : see this blog. Remark : A CAT(0) space is uniquely geodesic, but the converse is ...
0
votes
0answers
167 views

Proper Group action on a metric space

Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ ...
7
votes
4answers
372 views

isometric embeddings of Cayley graphs in “nice” spaces

This is from a physicist I know and as may be expected, I am threading my way between poorly defined and poorly translated. What groups have Cayley graphs (w.r.t. a fixed finite generating set, and ...
5
votes
1answer
235 views

Flat sector in a proper cocompact CAT(0) space

Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 ...
4
votes
1answer
294 views

Examples of CAT(0)-groups

My question is the following: Let M be a simply connected Riemannian manifold whose sectional curvatures are all nonpositive and let G be a group. Suppose that G acts in M properly discontinuous and ...
5
votes
2answers
674 views

Quasi-isometries vs Cayley Graphs

The following questions might be trivial, however, I couldn't solve them: Let $G$ be generated by a finite symmetric set $S.$ Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of ...