Timeline for Structure of the group generated by two specific symplectic matrices
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 7, 2016 at 9:53 | vote | accept | Matheus | ||
| Oct 31, 2015 at 10:52 | comment | added | Geoff Robinson | @Stefan: Thanks, yes I saw that it meant that $p$ is not faithful, I was just trying to reconcile your conclusion with what Matheus had said about words of length < 25. | |
| Oct 31, 2015 at 10:42 | comment | added | Matheus | @StefanKohl The fact that $p$ is not faithful rules out the strategy pointed out in my post and in Igor Rivin's comments to show the thinness of $<A,B>$, but, if I understand it correctly, $<A,B>$ might be thin anyway. Thus, if you don't mind, I propose to wait for a couple of days for an eventual answer to the original thinness question in my post before I give you the corresponding credit by accepting your answer, what do you think? | |
| Oct 31, 2015 at 10:19 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
The word can be written as a third power.
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| Oct 31, 2015 at 10:17 | comment | added | Matheus | @StefanKohl Thank you! In fact, you're right: while I tested all words of length $\leq 23$, I did only a partial test for words of length 24... | |
| Oct 31, 2015 at 10:11 | comment | added | Stefan Kohl♦ | @GeoffRobinson: It says that the representation $p$ is not faithful; apparently Matheus did not check all words of length $< 25$ in $A$, $B$, $A^2 = A^{-1}$ and $B^2 = B^{-1}$, unless he counted $A^2$ and $B^2$ as having length $2$. I found the word with GAP; actually I computed spheres of radii $1, 2 \dots$ about $1$, and noticed that sphere sizes dropped below $2^{r+1}$ at $r=12$. This then led to a nontrivial relation of length $2 \cdot 12 = 24$. | |
| Oct 31, 2015 at 10:00 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Write out explicitly an element of the kernel of p.
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| Oct 31, 2015 at 0:55 | comment | added | Geoff Robinson | This seems to be a non-trivial word of length 24. This doesn't actually contradict what the OP says, but I would like to understand what is going on. Presumably you produced this word with some computer algebra system? And did the OP check all words of length < 25 in $A,B,A^{2},B^{2}$? | |
| Oct 30, 2015 at 23:37 | history | answered | Stefan Kohl♦ | CC BY-SA 3.0 |