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

TL;DR

This paper constructs examples of finitely generated subgroups within certain lattice products that have positive first Betti number and whose faithful representations are quasi-isometric embeddings, challenging existing conjectures.

Contribution

It provides new examples of subgroups with positive Betti number in lattice products, demonstrating their quasi-isometric embedding properties into real semisimple Lie groups.

Findings

Existence of finitely generated subgroups with positive Betti number

Faithful representations are quasi-isometric embeddings

Counterexamples inspired by Bass-Lubotzky's work

Abstract

Let $\Delta$ be a cocompact lattice in $\mathsf{Sp}(m,1)$, $m \geq 2$, 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.