Timeline for How is this group theoretic construct called?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| May 23, 2019 at 14:52 | history | edited | user6671 | CC BY-SA 4.0 |
added 92 characters in body
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| May 23, 2019 at 13:42 | history | edited | user6671 | CC BY-SA 4.0 |
added 221 characters in body
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| May 23, 2019 at 13:37 | comment | added | user6671 | @AshotMinasyan: Ok thanks! That makes sense! | |
| May 23, 2019 at 13:35 | comment | added | Ashot Minasyan | @orgesleka: your map $\beta$ should go in the opposite direction, from the genuine direct product to your construction. The formula is the same, and it is a group isomorphism, as you can check directly. | |
| May 23, 2019 at 13:28 | comment | added | user6671 | @AshotMinasyan: Ok, I see that this defines an embedding from $G$ to $\mathbb{Z}\times_S G$ and this gives a bijection $\beta: \mathbb{Z} \times_S G \rightarrow \mathbb{Z} \times G, (a,g) \mapsto (a-|g|,g)$. But is this a homomorphism of groups? I don't see that. | |
| May 23, 2019 at 13:11 | comment | added | Ashot Minasyan | @orgesleka: Sorry, it should be $g \mapsto (-|g|,g)$. | |
| May 23, 2019 at 13:03 | comment | added | user6671 | @ashotminasyan: I don't understand how this is a group homomorphism? | |
| May 23, 2019 at 12:45 | comment | added | Ashot Minasyan | Since your $2$-cocycle is defined as the coboundary of the $1$-cochain $g \mapsto |g|$, the group that you get is the genuine direct product $\mathbb{Z} \times G$. In fact, the map $g \mapsto (|g|,g)$ defines an embedding of $G$ into your construction, which commutes with the natural copy of $\mathbb{Z}$. | |
| May 23, 2019 at 6:42 | history | edited | YCor | CC BY-SA 4.0 |
fixed typos
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| May 23, 2019 at 5:35 | history | edited | user6671 | CC BY-SA 4.0 |
added 149 characters in body
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| May 20, 2019 at 13:57 | comment | added | Misha | Sorry, it is $\psi(g,h^{-1})$, and assuming symmetric generating set so that $|h|=|h^{-1}|$. | |
| May 20, 2019 at 5:39 | comment | added | user6671 | @Misha: I don't think they are the same, although maybe related. | |
| May 19, 2019 at 21:28 | comment | added | Misha | $\psi(g,h)$ is twice the Gromov product. | |
| May 19, 2019 at 18:19 | comment | added | user6671 | @Misha: Thanks for your comment. Which quantity do you mean? | |
| May 19, 2019 at 18:12 | comment | added | Misha | If you divide this quantity by 2, it becomes Gromov product, denoted $(g,h)_e$, which is defined for general metric spaces, not just for Cayley graphs. | |
| May 19, 2019 at 11:09 | history | edited | user6671 | CC BY-SA 4.0 |
corrected error in definition of metric
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| May 18, 2019 at 8:06 | history | asked | user6671 | CC BY-SA 4.0 |