Fiber products of rank 1 superrigid lattices and quasi-isometric embeddings

Let $\mathrm{\Delta }$ be a cocompact lattice in $\mathsf{S}\mathsf{p}(m,1)$, $m\ge 2$, or $F_4^{(-20)}$. We exhibit examples of finitely generated subgroups of $\Delta \times \Delta$ with positive first Betti number all of whose discrete faithful representations into any real semisimple Lie group are quasi-isometric embeddings. The examples of this paper are inspired by the counterexamples of Bass-Lubotzky [BL00] to Platonov’s conjecture.