Timeline for Quotients of f.p. amenable groups
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 7, 2011 at 3:11 | comment | added | user6976 | By the way, it would be interesting to know what kind of finitely presented groups can cover infinitely presented metabelian groups. It may be that these finitely presented groups satisfy much stronger hyperbolicity properties than just "contain a free non-Abelian subgroup". Of course they are not necessarily hyperbolic because they can be direct products, for example. But still ... . Somebody may want to analyse the old result of Bieri and Strebel again. | |
| Dec 7, 2011 at 3:06 | comment | added | user6976 | @Mustafa: indeed, I forgot about that result of Bieri-Strebel. | |
| Dec 7, 2011 at 2:49 | comment | added | Benjamin Steinberg | Thanks. This makes the motivation for the question clearer. | |
| Dec 7, 2011 at 2:02 | comment | added | Mustafa Gokhan Benli | @Benjamin There is a theorem of Bieri and Strebel which says: If G is an infinitely presented metabelian group then any finitely presented covering group contains a nonabelian free group. | |
| Dec 7, 2011 at 1:12 | comment | added | user6976 | Take a finite number of relations $R$ of the lamplighter group $G$. To deduce these relations one needs, say, first $n$ of the standard relations $U$ of $G$. Therefore the group $H$ given by the relations $R \cup U$ is also given by $U$. Hence $H$ is a homomorphic image of the group $T$ given by $R$. But it is well known that $H$ is virtually free and non-amenable. Hence $T$ is not amenable either. Thus $G$ is not a homomorphic image of a finitely presented amenable group. The same works for the Grigorchuk group and for our lacunary hyperbolic group. | |
| Dec 7, 2011 at 0:56 | comment | added | Benjamin Steinberg | It was the lamplighter and Grigorchuk groups I was thinking of. | |
| Dec 7, 2011 at 0:38 | comment | added | user6976 | @Ben: In our paper "Lacunary hyperbolic groups" with Olshanskii and Osin, we construct a finitely generated amenable group which is a limit of hyperbolic groups. It could be that this group is not a homomorphic image of a finitely presented amenable group. Other possible examples are the lamplighter group (also a limit of hyperbolic groups) and Grigorchuk groups (a limit of virtual products of free groups). I do not know for sure, though. | |
| Dec 6, 2011 at 23:58 | comment | added | Benjamin Steinberg | Mark, are there any finitely generated amenable groups which are not quotients of finitely presented amenable groups? | |
| Dec 6, 2011 at 23:07 | comment | added | user6976 | No, you are right: she constructed the group in 1980. | |
| Dec 6, 2011 at 22:55 | comment | added | user6976 | I am not sure it was the very first example. Olga Kh.'s first example was also obtained in 1977 as far as I remember. In 1977 she got the Soviet Academy of Sciences "gold" prize for her group (although that might have happen in 1978). | |
| Dec 6, 2011 at 22:07 | comment | added | YCor | Indeed, Abels's group was the first example of a f.p. solvable group without max-n (i.e. with an ascending sequence of normal subgroups, or equivalently with an infinitely presented quotient), solving a question of P. Hall from the fifties. I guess it's also the first example of a f.p. amenable group without max-n. | |
| Dec 6, 2011 at 20:50 | vote | accept | Mustafa Gokhan Benli | ||
| Dec 6, 2011 at 20:49 | vote | accept | Mustafa Gokhan Benli | ||
| Dec 6, 2011 at 20:49 | |||||
| Dec 6, 2011 at 20:29 | history | answered | user6976 | CC BY-SA 3.0 |