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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 $\operatorname{CAT}(0)$?

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    $\begingroup$ 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)? $\endgroup$
    – YCor
    Commented Apr 21, 2021 at 9:11

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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.

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    $\begingroup$ I do know that it is an open question whether hyperbolic groups are CAT(0). I am interested in the case of more than one factor: even though hyperbolic groups may be wild, the only examples I know of groups acting geometrically on products of hyperbolic spaces are rather "nice". $\endgroup$
    – Thomas Haettel
    Commented Apr 21, 2021 at 11:51
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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.

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