Quantifying separability in limit groups via representations

We show that for any finitely generated subgroup H of a limit group L there exists a finite-index subgroup K containing H, such that K is a subgroup of a group obtained from H by a series of extensions of centralizers and free products with Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}$$\end{document}. If H is non-abelian, the K is fully residually H. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology.rollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes the results in (Louder et al. in Sel Math New Ser 23:2019-2027, 2017). Another corollary is that a hyperbolic limit group satisfies the Geometric Hanna Neumann conjecture.