Timeline for For a group with one end does the property of connected spheres follow?
Current License: CC BY-SA 3.0
15 events
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| Jul 14, 2012 at 10:41 | comment | added | Olga | Thank you very much for your valuable comments - if the property holds for at least the finitely presented groups, it's already very well. | |
| Jul 4, 2012 at 18:03 | comment | added | Misha | ... coarsely C-connected, then coarse M-V theorem would imply that the loop in question bounds a cycle in a $R(C)$-Rips complex contradicting the assumption. I am not sure what happens with rings other than Z. | |
| Jul 4, 2012 at 18:00 | comment | added | Misha | I think one can also prove this property for groups which are $FP_2$ over a commutative unital ring, e.g. integers. On the other hand, all groups which are not $FP_2$ over $Z$ proobably provide countere-examples via a coarse form of Meyer-Vietoris theorem, where you split the group over a large sphere crossing a loop in the Cayley graph which does not bound a cycle in a D-Rips complex with some reasonably large D. The point is that both metric ball bounded by this sphere and it's complement are coarsely c- connected for some c which depends only on the group. If the sphere .... | |
| Jul 4, 2012 at 12:58 | comment | added | Valerio Capraro | @Lee: indeed that's exactly Antoine's proof on my blackboard. | |
| Jul 4, 2012 at 12:52 | comment | added | Lee Mosher | Previous comment cross-posted with Valerio. | |
| Jul 4, 2012 at 12:51 | comment | added | Lee Mosher | @Ashot: you are correct that this is true when $G$ is finitely presented. In the Cayley graph $\Gamma$ for $G$, if the "property of connected spheres" fails then for each $C$ there exists $R$ for which the set $(B_R)^c_\infty \cap B_{R+C}$ is disconnected; letting $P = \frac{R+C}{2}$ one can show that the 2-complex obtained from $\Gamma$ by attaching discs to all closed edge paths of length $\le P$ is not simply connected. Since $C$ can be arbitrarily large, $P$ is also arbitrarily large, so $G$ is not finitely presented. If this is useful I'll put more details in an answer, but I'm out of roo | |
| Jul 4, 2012 at 12:47 | comment | added | Valerio Capraro | I am sorry, who's this friend? Antoine Gournay put yesterday a paper in arxiv where he uses basically the same property. He has a sketch of proof that the property holds for all finitely presented group. We have had a chat right now and we are pretty convinced that it fails for every one-ended group with at least one non simply connected asymptotic cone. | |
| Jul 4, 2012 at 11:34 | comment | added | Ashot Minasyan | My guess is that this might be true if $G$ is finitely presented, but should be false for finitely generated groups in general. Indeed, there are f.g. groups with one end that are limits (in the space of marked groups) of f.p. groups, each of which has infinitely many ends. One such example is the lamplighter group $(\mathbb{Z}/2\mathbb{Z}) wr \mathbb{Z}$. This is the group I would try to look at first. | |
| Jul 4, 2012 at 7:53 | comment | added | Olga | Thank you very much for your comment: for sure, $z_k$ have to lie in the fiber considered. Secondly, what is important, a case of dead ends is not an issue here - I corrected a definition of connected spheres after your remark, see a new one above. | |
| Jul 4, 2012 at 7:50 | history | edited | Olga | CC BY-SA 3.0 |
corrected the definition of a property of connected spheres after the remark of M. Sapir; added 17 characters in body
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| Jul 4, 2012 at 7:34 | history | edited | Olga | CC BY-SA 3.0 |
added 61 characters in body
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| Jul 4, 2012 at 6:36 | comment | added | user6976 | The question does not make sense for me. For every two elements $x,y$ there exists a path in the left Cayley graph from $x$ to $y$. The label of that path is your $g$. Perhaps you wanted all elements $z_k=g_{k+1}...g_ny$ to satisfy $R<|z_k|<R+C$? In that case the answer is "no" because of "dead ends of arbitrary depth". | |
| Jul 4, 2012 at 6:31 | history | edited | Olga | CC BY-SA 3.0 |
given a definition of a group with one end; added 1 characters in body
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| Jul 4, 2012 at 6:21 | history | edited | Olga | CC BY-SA 3.0 |
corrected spelling; added 2 characters in body
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| Jul 4, 2012 at 5:23 | history | asked | Olga | CC BY-SA 3.0 |