Cartan projections of fiber products and non quasi-isometric embeddings

Several important classes of groups are known to admit discrete faithful linear representations which are quasi-isometric embeddings. These include Anosov hyperbolic groups and convex cocompact subgroups of the projective linear group $\mathrm{PSL}(d,\mathbb{R})$. In this talk, we will be interested in groups with the opposite behaviour. We are going to exhibit some constructions of linear finitely generated (and presented) subgroups $P$ of direct products of hyperbolic groups, with the property that every linear representation of $P$ (over a local field) cannot be a quasi-isometric embedding. The main tool for the proof is an upper estimate for the norm of the Cartan projection of multiple commutators in direct products.