Cartan projections of fiber products and non-quasi-isometric embeddings

Let $\Gamma$ be a finitely generated group and $N$ be a normal subgroup of $\Gamma$. The fiber product of $\Gamma$ with respect to $N$ is the subgroup $\Gamma {\times }_N\Gamma =\{(\gamma ,\gamma w):\gamma \in \Gamma ,w\in N\}$ of the direct product $\Gamma \times \Gamma$. For every representation $\rho :\Gamma {\times }_N\Gamma \to {\mathrm{GL}}_d\left(k\right)$, where $k$ is a local field, we establish upper bounds for the norm of the Cartan projection of $\rho$ in terms of a fixed word length function on $\Gamma$. As an application, we exhibit examples of finitely generated and finitely presented fiber products $P=\Gamma {\times }_N\Gamma$, where $\Gamma$ is linear and Gromov hyperbolic, such that $P$ does not admit linear representations that are quasi-isometric embeddings.