The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then it could be rather hard; I'm asking this question here because MO is lucky enough to have some of the foremost experts on lacunary hyperbolic groups as active participants.


**Definitions**


1. A group $\Gamma$ is *non-Hopfian* if there is an epimorphism $\Gamma\to\Gamma$ with non-trivial kernel.
2. A group $\Gamma$ is *lacunary hyperbolic* if some asymptotic cone of $\Gamma$ is an $\mathbb{R}$-tree.


To motivate this second definition, note that a group is word-hyperbolic if and only if *every* asymptotic cone is an $\mathbb{R}$-tree.


Lacunary hyperbolic groups were defined and investigated in a paper of [Ol'shanskii, Osin and Sapir](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=sapir&s5=osin&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) (although examples of lacunary hyperbolic groups that are not hyperbolic already existed---I believe the first one was constructed by Simon Thomas). They construct examples that exhibit very non-hyperbolic behaviour, including torsion groups and Tarski monsters.


**Question**


Once again:-



> 
> Is there a non-Hopfian lacunary hyperbolic group?
> 
> 
> 


**Motivation**


My motivation comes from logic, and the following fact.


**Proposition:** A lacunary hyperbolic group is a direct limit of hyperbolic groups (satisfying a certain injectivity-radius condition).


That is to say, lacunary hyperbolic groups are limit groups over the class of all hyperbolic groups. (Note: the injectivity-radius condition means that there are other limit groups over hyperbolic groups which are not lacunary hyperbolic. I'm also interested in them.) Sela has shown that limit groups over a *fixed* hyperbolic group $\Gamma$ (and its subgroups) tell you a lot about the solutions to equations over $\Gamma$. For instance, his result that a sequence of epimorphisms of $\Gamma$-limit groups eventually stabilises (which implies that all $\Gamma$-limit groups are Hopfian) has the following consequence.


**Theorem (Sela):** Hyperbolic groups are equationally Noetherian. That is, any infinite set of equations is equivalent to a finite subsystem.


In the wake of Sela's work we have a fairly detailed understanding of solutions to equations over a given word-hyperbolic group $\Gamma$. But it's still a matter of great interest to try to understand systems of equations over *all* hyperbolic groups.


Pathological behaviour in lacunary hyperbolic groups should translate into pathological results about systems of equations over hyperbolic groups. A positive answer to my question would imply that the class of hyperbolic groups is not equationally Noetherian. And that would be quite interesting.


**Note:** [This paper](http://arxiv.org/abs/0903.3978) of Denis Osin already makes a connection between equations over a single lacunary hyperbolic group and equations over the class of all hyperbolic groups.




---


This question attracted great answers from Yves Cornulier and Mark Sapir, as well as some excellent comments from Denis Osin. Let me quickly clarify my goals in answering the question, and try to summarise what the state of knowledge seems to be. I hope someone will correct me if I make any unwarranted conjectures!


My motivation came from the theory of equations over the class of all (word-)hyperbolic groups. For these purposes, it is not important to actually find a non-Hopfian lacunary hyperbolic group; merely a non-Hopfian limit of hyperbolic groups is enough. (That is, the injectivity radius condition in the above proposition can be ignored.) Yves Cornulier gave an example of a limit of virtually free (in particular, hyperbolic) groups which is non-Hopfian. From this one can conclude that the class of word-hyperbolic groups is not equationally Noetherian, as I had hoped.


[Note: I chose to accept Yves's answer. Mark's answer is equally worthy of acceptance.]


Clearly, the pathologies of Yves's groups derive from torsion---the class of free groups is equationally Noetherian---and there are some reasons to expect torsion to cause problems, so I asked in a comment for torsion-free examples. These were provided by Denis Osin, who referred to a [paper](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=ivanov&s5=storozhev&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=2174100) of Ivanov and Storozhev. Thus, we also have that the class of all torsion-free hyperbolic groups is not equationally Noetherian.


Let us now turn to the question in the title---what if we require an actual lacunary hyperbolic group that is non-Hopfian. First, it seems very likely that such a thing exists. As Mark says, 'A short answer is "why not?"'. More formally, Denis claims in a comment that the subspace of the space of marked groups consisting of lacunary hyperbolic groups is comeagre in the closure of the subspace of hyperbolic groups. This formalises the idea that lacunary hyperbolic groups are not particularly special among limits of hyperbolic groups.


Mark also suggested two possible approaches to constructing a non-Hopfian lacunary hyperbolic group; however, in a comment, Denis questioned whether one of these approaches works. In summary, I feel fairly confident in concluding that a construction of a non-Hopfian lacunary hyperbolic group is not currently known, although one should expect to be able to find one with a bit of work.



Is there a non-Hopfian lacunary hyperbolic group?
Contrary to I think the expectations of the earlier answers, in [a recent paper](http://arxiv.org/abs/1606.00679), Coulon--Guirardel showed that the answer to the title question is 'no'!



> 
> **Theorem (Coulon--Guirardel):** Every lacunary hyperbolic group is Hopfian.
> 
> 
> 


The proof is remarkably short, and although it of course uses the fact that hyperbolic groups are Hopfian, it doesn't need the details of the proof.


