Timeline for When is an HNN-extension finitely presented?
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22 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2021 at 9:53 | vote | accept | ADL | ||
| Aug 15, 2012 at 19:24 | answer | added | user6976 | timeline score: 8 | |
| Aug 13, 2012 at 9:51 | comment | added | HJRW | Thanks for the interesting comments, Mark. Perhaps either you or Benjamin could write an answer about L-presentations? | |
| Aug 12, 2012 at 1:56 | comment | added | user6976 | More examples can be found in Sapir, Mark, Wise, Daniel T., Ascending HNN extensions of residually finite groups can be non-Hopfian and can have very few finite quotients. J. Pure Appl. Algebra 166 (2002), no. 1-2, 191–202 and in Olʹshanskii, Alexander Yu.; Sapir, Mark V. Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Études Sci. No. 96 (2002), 43–169. | |
| Aug 12, 2012 at 1:54 | comment | added | user6976 | For ascending HNN extensions, i.e. $H=K$ (as in Baumslag-Remeslennikov case, in the Grigorchuk case, and many others) one needs, as Ben Steinberg ponted out that $H$ has a finite L-presentation with respect to the endomorphism $\phi:H\to K'$. That is there are finite number of relations $r_1=1,...,r_k=1$ so that the set of relations $\\{\phi^m(r_j)=1\mid m\ge 0, 1\le j\le k\\}$ defines $H$. | |
| Aug 12, 2012 at 1:46 | comment | added | user6976 | @Henry: This is a group constructed by Baumslag and Remeslennikov (independently). In fact they had a more general construction (with $t-1$ replaced by any monic polynomial with at least two monomials). The Baumslag-Gersten group is $\langle a,b\mid a^{a^b}=a^2\rangle$. It was also constructed by Baumslag, and Gersten had no role in it. He did use it and proved some properties of it. | |
| Aug 10, 2012 at 15:46 | comment | added | Benjamin Steinberg | Finite L-presentations let you get finitely presented HNN extensions in many cases. | |
| Aug 10, 2012 at 13:20 | history | edited | ADL | CC BY-SA 3.0 |
edited body
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| Aug 10, 2012 at 13:18 | comment | added | HJRW | AL - thanks for the edit. I think you mean 'If you put these restrictions on $H$ and $K$...', don't you? | |
| Aug 10, 2012 at 13:17 | comment | added | HJRW | Yves - many thanks! That's very interesting. (I think the Baumslag's group I mentioned used to be called 'the Baumslag--Gersten group'. Somehow Gersten got forgotten.) | |
| Aug 10, 2012 at 13:01 | history | edited | ADL | CC BY-SA 3.0 |
Clarifying things a bit, as discussed in the comments.
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| Aug 10, 2012 at 12:53 | comment | added | YCor | @HW: I forgot to say: the above endomorphism of $Z[t,t^{-1}]$ is first extended to the semidirect product, acting trivially on $t$. This gives an injective endomorphism of ZwrZ, out of which the ascending HNN extension is made. | |
| Aug 10, 2012 at 12:47 | comment | added | YCor | @HW: there are several Baumslag groups :p... view ZwrZ as the semidirect product $Z[t,t^{-1}]\rtimes \langle t\rangle$. (this is certainly not Baumslag's language, who rather likes group presentations). Consider the injective homomorphism of the free abelian group of infinite rank $Z[t,t^{-1}]$ given by multiplication by $(t-1)$. The associated ascending HNN-extension is thus the semidirect product $Z[t,t^{-1},(t-1)^{-1}]\rtimes \langle t,t-1\rangle$, where $\langle t,t-1\rangle$ is isomorphic to $Z^2$. Baumslag proved it's f.p., with a more conceptual and general proof by Bieri-Strebel. | |
| Aug 10, 2012 at 12:17 | comment | added | HJRW | AL - thanks, that considerably clarifies things. You might want to edit the question accordingly. | |
| Aug 10, 2012 at 12:05 | comment | added | ADL | @HW: The sort of answer I would hope for would be "If you put these restrictions on $G$ and $K$ (and perhaps on the isomorphism between $K$ and $K^{\prime}$) then you can say something." Along the lines of the Baumslag-Tretfoff conditions for residual finiteness; sufficient, but not necessary. | |
| Aug 10, 2012 at 11:59 | comment | added | HJRW | Yves - huh, I didn't know that. Which Baumslag's group? $\langle a,b\mid a^{(a^b)}=a^2\rangle$? Steve D - sorry if so! | |
| Aug 10, 2012 at 11:47 | comment | added | YCor | @HW: probably SteveD means that ZwrZ admits a f.p. HNN extension (Baumslag's group). | |
| Aug 10, 2012 at 11:43 | comment | added | HJRW | Steve D - $\mathbb{Z}\wr\mathbb{Z}$ is not finitely presented! | |
| Aug 10, 2012 at 11:43 | comment | added | HJRW | What sort of answer do you (the OP) hope for from this question? As you correctly point out, the naive guess is false. As Yves says, it's unlikely that there is an answer. | |
| Aug 10, 2012 at 11:41 | comment | added | Steve D | A similar example to the one by Yves is the wreath product $\mathbb{Z}\wr\mathbb{Z}$. | |
| Aug 10, 2012 at 11:27 | comment | added | YCor | Well, as you justify, the answer is not obvious. The problem is non-trivial (and likely has no general answer) even it the most degenerate instance of HNN, namely that of a semidirect product $G=H\rtimes\mathbf{Z}$. This group can be finitely presented even if $H$ is not finitely presented, or even finitely generated. For instance, the Thompson group $F$ of the interval can be written in either of these ways (and also with $H$ finitely presented). Such phenomena also occur in the context of metabelian groups. | |
| Aug 10, 2012 at 10:50 | history | asked | ADL | CC BY-SA 3.0 |