Weakly positive and directed Anosov representations

Given a finitely generated group $Γ$, a directed graph $Λ$, and a map $R:Λ\toΓ$, we introduce the notion of an $(R,Λ)$-directed Anosov representation. This is a weakening of the notion of Anosov representations. Our main theorem gives a procedure to construct $(R,Λ)$-directed Anosov representations using Fock-Goncharov positivity. As an application of our main theorem, we construct large families of primitive stable representations from $F_2$ to $\mathrm{PGL}(V)$, including non-discrete and non-faithful examples.

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