Timeline for "Quasi-orthogonal" subgroup of a group with length?
Current License: CC BY-SA 3.0
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| Oct 20, 2015 at 13:11 | comment | added | Kolya Ivankov | Well, I think Your condition is stronger, and actually, after reconsidering, it was the one I came up with some time ago and them forgotten - so thank You. I also think that this is satisfied if the group is hyperbolic, and I am also not sure if it's not just a trivial consequence from the definition of the length. I haven't been working with the properties of groups which were less trivial than just compactness or commutativity, therefore the question. | |
| Oct 20, 2015 at 12:27 | comment | added | Vincent | I haven't seen it before so I cannot answer your question, but I still find it interesting. Here is another question. What about the weaker following condition: "Let $g \in G$ and let $h' = \rho(g)^{-1}g \in H$. The weaker requirement now is: The length $|h'|$ depends only on $|g|$, and the dependence is at most polynomial."? This would sound more natural to me. Do you know if this weak condition has a name? Or does it follow directly from the definition of length and hence doesn't need a name? | |
| Oct 20, 2015 at 7:24 | comment | added | Kolya Ivankov | Well, as to the term, this is the best one I can come up with, out of purely geometric picturing of the group as some kind of "orthogonal sum" of the coset set and a subgroup. As to your second question - sorry, never thought of it. My closest examples are the groups with finite subgroups, central subgroups of finite index, and several more, like $\mathbb{Z}/2\mathbb{Z}$ as a subgroup of the fundamental groups of n-cusp space. | |
| Oct 20, 2015 at 7:01 | comment | added | Vincent | Why do you use the term 'Quasi-orthogonal'? Do orthogonal groups viewed as subgroups of general linear groups satisfy a stronger version of this property? If so, what is it? | |
| Oct 20, 2015 at 6:29 | history | edited | Kolya Ivankov | CC BY-SA 3.0 |
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| Oct 20, 2015 at 6:24 | history | edited | Kolya Ivankov | CC BY-SA 3.0 |
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| Oct 20, 2015 at 6:14 | history | asked | Kolya Ivankov | CC BY-SA 3.0 |