Timeline for Is there a non-Hopfian lacunary hyperbolic group?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 28, 2011 at 10:40 | vote | accept | HJRW | ||
| Jun 16, 2016 at 10:37 | |||||
| Sep 21, 2011 at 18:42 | comment | added | Denis Osin | Henry, yes, of course you are right, $\mathcal H_n$ is the closure of the set of hyperbolic groups. | |
| Sep 21, 2011 at 7:07 | comment | added | HJRW | Denis - thanks for your extremely useful comments. It's a little bit of a shame that they're not posted as an answer, as I think they deserve some credit! Is $\mathcal{H}_n$ meant to be the closure of the set of hyperbolic groups? | |
| Sep 21, 2011 at 6:05 | comment | added | Denis Osin | @Henry, continuation: 2. In a sense, lacunary hyperbolic groups are generic. It is not hard to show the following. Fix an integer $n>1$ and let $\mathcal H_n$ (respectively, $\mathcal L_n$ be the subspace of presentations of non-elementary hyperbolic (respectively, lacunary hyperbolic) groups in the space of marked group presentations with $n$ generators. Then $\mathcal L_n$ is a dense $G_\delta $ subset of $\mathcal H_n$. In particular, it is comeagre. | |
| Sep 21, 2011 at 5:57 | comment | added | Denis Osin | @Henry: This is the answer to the follow up questions. 1. Such groups exist and are constructed by Ivanov and Storozhev in arXiv:math/0312491. These groups are limits of groups $G(i)$, which are hyperbolic by Lemma 1 from the paper. (The fact that condition R implies hyperbolicity can be easily extracted from Olshanskii's book cited in the paper). And the groups $G(i)$ are torsion free by Lemma 2. | |
| Sep 20, 2011 at 9:36 | comment | added | HJRW | ... I have two follow-up questions. 1. It would be even more interesting to find a non-Hopfian group which was a limit of torsion-free hyperbolic groups. Do you know of such an example? 2. Can anything be said about the set of lacunary hyperbolic groups inside the set of all limits of hyperbolic groups, which would perhaps make clear how special lacunary hyperbolic groups are? | |
| Sep 20, 2011 at 9:33 | comment | added | HJRW | Many thanks for your answer, Yves. My question focussed on lacunary hyperbolic groups because they seem well studies and of interest in their own right; but your answer is good enough for the application I have in mind, and has helped me a lot in appreciating the difference between lacunary hyperbolic groups and more general limits of hyperbolic groups. | |
| Sep 19, 2011 at 6:17 | comment | added | YCor | @Mark: $Z(B)$ is the set of $3\times 3$ matrices with 1 in the diagonal, $a_{12}=a_{23}=0$ and $a_{13}\in B$. Besides, if you pick "having a linear Dehn function" as a definition of being hyperbolic, C'(1/6) small cancelation groups are hyperbolic and this is due to Greendlinger in the 50s. This is before Novikov-Adian and is also less hard, and limits of C'(1/6)-groups are a rich source of infinitely presented limits of hyperbolic groups. | |
| Sep 19, 2011 at 3:38 | comment | added | user6976 | @Yves: $B$ is a ring? What is $Z(B)$? As you said, many non-lacunary hyperbolic groups are limits of hyperbolic groups. Historically the first one was the infinite Burnside group $B_{m,n}$ from the paper of Novikov and Adian. The fact that the intermediate groups in their construction are hyperbolic is proved in their paper (many years before the concept of hyperbolic group was introduced). Also the lamplighter group is a limit of virtually free groups, etc. Groups with a non-trivial law or with a central element of infinite order cannot be lacunary hyperbolic by our result with Drutu. | |
| Sep 18, 2011 at 20:52 | history | answered | YCor | CC BY-SA 3.0 |