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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$\Gamma \times_N \Gamma=\big \{(\gamma, \gamma w): \gamma \in \Gamma, w \inN\big \}$ of the direct product $\Gamma \times \Gamma$. For everyrepresentation $\rho:\Gamma \times_N \Gamma \rightarrow \mathsf{GL}_d(k)$,where $k$ is a local field, we establish upper bounds for the norm of theCartan projection of $\rho$ in terms of a fixed word length function on$\Gamma$. As an application, we exhibit examples of finitely generated andfinitely presented fiber products $P=\Gamma \times_N \Gamma$, where $\Gamma$ islinear and Gromov hyperbolic, such that $P$ does not admit linearrepresentations which are quasi-isometric embeddings.

Cartan projections of fiber products and non quasi-isometric embeddings