Recall that a group $G$ is called Hopfian if every surjective endomorphism $G\to G$ is injective. Malcev observed that all finitely-generated (f.g.) residually finite groups are Hopfian. It is well-known that residual finiteness is not a coarse invariant, i.e. a residually finite f.g. group can be quasi-isometric to a non-residually finite one. For instance, Burger and Mozes proved that $F_2\times F_2$ is quasi-isometric to a simple group. Earlier examples, due to Deligne, were of non-residually finite central extensions of residually finite groups, with kernel of order 2. Deligne's examples imply that residual finiteness is not even a virtual isomorphism invariant.
Question 1: Is Hopfian property of preserved by quasi-isometries of f.g. groups?
Natural candidates would be examples of non-Hopfian CAT(0) groups constructed by Dani Wise in his thesis. However, I do not know if such groups are quasi-isometric to, say, residually finite groups. A subquestion of Question 1 is:
Question 2. Suppose that $G$ is a group acting geometrically on a product of simplicial trees of finite valence. Can $G$ be non-Hopfian?
Note that such $G$ is necessarily quasi-isometric to a product of free groups and such products are residually finite.