Quasi-finiteness of morphisms between character varieties

Let $f: Y\to X$ be a morphism between smooth complex quasi-projective varieties and $Z$ be the closure of $f(Y)$ with $ι: Z\to X$ the inclusion map. We prove that a. for any field $K$, there exist finitely many semisimple representations $\{τ_i:π_1(Z)\to {\rm GL}_N(\overline{k})\}_{i=1,\ldots,\ell}$ with $k\subset K$ the minimal field contained in $K$ such that if $\varrho:π_1(X)\to {\rm GL}_{N}(K)$ is any representation satisfying $[f^*\varrho]=1$, then $[ι^*\varrho]=[τ_i]$ for some $i$. b. The induced morphism between ${\rm GL}_{N}$-character varieties (of any characteristic) of $π_1(X)$ and $π_1(Y)$ is quasi-finite if ${\rm Im}[π_1(Z)\to π_1(X)]$ is a finite index subgroup of $π_1(X)$. These results extend the main results by Lasell in 1995 and Lasell-Ramachandran in 1996 from smooth complex projective varieties to quasi-projective cases with richer structures.