Cartan projections of fiber products and non‐quasi‐isometric embeddings.

Let Γ$\Gamma$ be a finitely generated group and N$N$ be a normal subgroup of Γ$\Gamma$. The fiber product of Γ$\Gamma$ with respect to N$N$ is the subgroup Γ×NΓ={(γ,γw):γ∈Γ,w∈N}$\Gamma \times _N \Gamma =\big \lbrace (\gamma, \gamma w): \gamma \in \Gamma, w \in N\big \rbrace$ of the direct product Γ×Γ$\Gamma \times \Gamma$. For every representation ρ:Γ×NΓ→GLd(k)$\rho:\Gamma \times _N \Gamma \rightarrow \mathsf {GL}_d(k)$, where k$k$ is a local field, we establish upper bounds for the norm of the Cartan projection of ρ$\rho$ in terms of a fixed word length function on Γ$\Gamma$. As an application, we exhibit examples of finitely generated and finitely presented fiber products P=Γ×NΓ$P=\Gamma \times _N \Gamma$, where Γ$\Gamma$ is linear and Gromov hyperbolic, such that P$P$ does not admit linear representations that are quasi‐isometric embeddings. [ABSTRACT FROM AUTHOR]