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 luck 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][1] (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][2] of Denis Osin already makes a connection between equations over a single lacunary hyperbolic group and equations over the class of all hyperbolic groups.


  [1]: 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
  [2]: http://arxiv.org/abs/0903.3978