Cartan projections of fiber products and non-quasi-isometric embeddings (Q6641579)

Let \(\textsf{GL}_d(k)\) be the linear group of matrices of dimension \(d\) on a local field \(k\). Let us consider the cone \(\textsf{E}^+:=(x_1, \dots, x_d)\in \mathbb{R}^d\ :\ x_1 \geq \cdots \geq x_n\}\) equipped with the standard Euclidean norm \(|| \cdot ||_{\mathbb{E}}\). Let \(\mu : \textsf{GL}_d(k) \rightarrow \textsf{E}^+\) be the Cartan projection. Let \(\Delta\) be a finitely generated group with a left invariant word metric induced by a finite generating subset of \(\Delta\) and denote by \(|\cdot |_{\Delta} : \Delta \rightarrow \mathbb{N}\) the associated word length function. A linear representation \(\psi: \Delta \rightarrow \textsf{GL}_d(k)\) is called a quasi-isometric embedding if the norm of the Cartan projection of \(\psi(\Delta)\) grows uniformly linearly in the word length on \(\Delta\).\N\NLet \(\Gamma\) 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 \(\rho : \Gamma \times_N \Gamma \rightarrow \textsf{GL}_d(k)\), the author establishes throughout the paper several upper bounds for the norm of the Cartan projection of \(\rho\) in terms of a fixed word length function on \(\Gamma\). He also provides many 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. An appendix is given on Distortion of fiber products into direct products and commensurability.