# Groups acting on products of hyperbolic spaces

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 $$\operatorname{CAT}(0)$$?

• An auxiliary question would be: if $X_i$ are proper spaces (say hyperbolic here), and $X=\prod_{i=1}^nX_i$ admits a proper cocompact action of a discrete group (i.e., $\mathrm{Isom}(X)$ has a cocompact lattice, does each $X_i$ admit such an action (i.e., does each $\mathrm{Isom}(X_i)$ admit a cocompact lattice)?
– YCor
Apr 21 at 9:11