TITLE: Groups acting on products of hyperbolic spaces
QUESTION [7 upvotes]: I am interested in groups acting properly and cocompactly by isometries on finite products of Gromov-hyperbolic metric spaces. I am mostly interested in the case where the group itself is not virtually a product of Gromov-hyperbolic groups.
Classical examples include irreducible uniform lattices in products or rank $1$ simple Lie groups, and irreducible uniform lattices acting on products of trees. These examples are all $\operatorname{CAT}(0)$.

Question. Are there examples of such groups which are not $\operatorname{CAT}(0)$, or not known to be $\operatorname{CAT}(0)$?

REPLY [3 votes]: This question for a product of one Gromov hyperbolic space is a famous open problem (see for example Bestvina's problem list) and as far as I am aware the general case is also open.

REPLY [3 votes]: Though it does not answer this question, the paper https://arxiv.org/abs/1509.03748 contains a weaker notion of nonpositive curvature than CAT(0) which all groups acting geometrically on product of Gromov hyperbolic groups satisfy and such that there are examples of non-CAT(0)-groups which also satisfy this weaker notion.