All Questions
Tagged with mg.metric-geometry geometric-group-theory
114 questions
4
votes
0
answers
98
views
Nice minimal embeddings of large finite groups into compact Riemannian manifolds
The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...
- 18.4k
2
votes
0
answers
241
views
Finitely generated groups non-embeddable into $L_1(0,1)$
I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group $\...
- 4,878
11
votes
1
answer
403
views
Embeddings of finitely generated groups into uniformly convex Banach spaces
de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...
- 4,878
6
votes
2
answers
729
views
Rationality of translation lengths in hyperbolic groups
Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...
- 619
18
votes
1
answer
502
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 ...
- 13.5k
6
votes
2
answers
323
views
Hilbert space compression of the lamplighter group
What is the Hilbert space compression exponent of the standard lamplighter group $\mathbb{Z_{2}} \wr \mathbb{Z}$? For $\mathbb{Z} \wr \mathbb{Z}$ it is known to be $2/3$ by work of Austin, Naor and ...
- 2,449
16
votes
1
answer
1k
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 ...
- 828
11
votes
0
answers
734
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 false (...
0
votes
1
answer
1k
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 $U=$ {...
- 1
9
votes
4
answers
982
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 ...
- 1,625
5
votes
1
answer
357
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 (3)...
- 53
6
votes
1
answer
768
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
...
- 235
10
votes
0
answers
458
views
is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
8
votes
2
answers
2k
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 $G$...
- 244