Timeline for Solution of an equation over free group
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 30, 2023 at 6:25 | comment | added | HJRW | @Shri: thanks, I understand now. | |
| Sep 30, 2023 at 5:28 | comment | added | Shri | @HJRW the word $w$ is coefficient also. The equation $w.w(t_1, \ldots, t_n) = 1$ is same as $w(t_1, \ldots, t_n) = w^{-1}$. | |
| Sep 29, 2023 at 15:32 | comment | added | HJRW | I don't understand the question. You can study equations over groups with or without coefficients. Since you say that the $t_i$ are all variables, your equation does not seem to have any coefficients. Such an equation always has a solution, namely the trivial one: setting $t_i=1$ for all $i$ gives the solution $w(1,\ldots,1)=1$. Please clarify: does your inclusion also include coefficients, are you hypothesising there are no non-trivial solutions, or something else? | |
| Sep 29, 2023 at 5:30 | comment | added | Shri | @SamNead Take $w =a^3[a,b]([a^{-1},b])^2$ in $F_2$ with basis $a, b$. As there is a finite group generated by 2 elements where the image of the word map $w$ is not closed with respect to inverse, hence the equation does not have any solution in free group. See this article's lemma 4.2. | |
| Sep 28, 2023 at 17:39 | comment | added | Sam Nead | Could you please give an example of a word $w$ as in your first paragraph? | |
| Sep 28, 2023 at 12:08 | history | edited | Shri | CC BY-SA 4.0 |
deleted 23 characters in body
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| Sep 28, 2023 at 11:15 | history | edited | Shri | CC BY-SA 4.0 |
Unsolved is changed to having no solution
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| Sep 28, 2023 at 11:12 | comment | added | Shri | @YCor Yes and thanks. | |
| Sep 28, 2023 at 10:35 | comment | added | YCor | Does "unsolved" mean "has no solution"? | |
| Sep 28, 2023 at 10:11 | history | asked | Shri | CC BY-SA 4.0 |