Timeline for Algorithm to calculate $Wh(G)$ for finitely presented group $G$?
Current License: CC BY-SA 3.0
8 events
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| Jun 25, 2011 at 16:59 | comment | added | Igor Belegradek | @Mark, your are right, now that I think about it $\mathrm{Wh}$ is always countable. Thanks for clarifying this and other things. | |
| Jun 25, 2011 at 15:11 | comment | added | user6976 | Since we can prove that an element is 1 if it is equal to 1, we can also prove that a f.p. group is trivial if it is trivial (there is a finite witness to this event). On the other hand I do not see how to prove that $Wh$ is trivial if it is known to be trivial or how to construct a nontrivial element in $Wh$ if it is known that $Wh$ is non-trivial. | |
| Jun 25, 2011 at 15:07 | comment | added | user6976 | @Igor: $Wh$ of a countable group cannot be uncountable. Right? The algorithm can in principle give elements of $Wh$ one by one as matrices over the group ring $\mathbb{Z}G$ (with integer coefficients). The problem is that it is not clear when a matrix is 1 in $Wh$. We actually can decide when a matrix is 1 (just represent it as a product of elementary matrices in a ring of matrices of some bigger size), but we cannot (I think) decide that the matrix is <b> not</b> 1. This is similar to deciding if a word is (is not) equal to 1 in a f.p. group. | |
| Jun 25, 2011 at 11:44 | comment | added | Igor Belegradek | @Mark, I do not have a firm grip what an algorithm is, but how could an uncountable abelian group, such as $\mathbb R$, be the output of an algorithm? What if the abelian group $\mathrm{Wh}(G)$ is uncountable? | |
| Jun 25, 2011 at 8:57 | comment | added | user6976 | @Igor: The question makes sense even if the $Wh$ is infinite. One can ask for an algorithm to list the elements of the group one by one. For example, the word problem is decidable in that sense in every finitely presented group because there exists an algorithm listing all words that are equal to 1. In that sense it is not clear if there exists an algorithm that lists all elements of $Wh$. At least I do not have any intuition what the answer should be. It is not even clear if there exists an algorithm that finds a non-trivial element of $Wh$ if there is one. | |
| Jun 23, 2011 at 17:09 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Jun 23, 2011 at 15:47 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Jun 23, 2011 at 15:26 | history | answered | Igor Belegradek | CC BY-SA 3.0 |