# Do quasi-isometries of semi direct products of free groups preserve fibres

Is it true that any quasi-isometry between semi direct products $Z^{n} \times_{f} F, Z^{n} \times_{f'} F$ for $F$ finitely generated free non-abelian group (for $F$ abelian it is obviously false) sends $Z^{n} \times \lbrace{f \rbrace}$ uniformly close to $Z^{n} \times \lbrace{f' \rbrace}$ for some $f' \in F$ and all $f \in F$? I've read in lecture notes - "Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups" that it is true for direct product $Z^{n} \times F$ and semi-direct products $Z^{n} \times_{f} F$ seem to be more dynamically complicated. Thank you for all your answers.

• Good question. The positive answer follows from: Lee Mosher, Michah Sageev, Kevin Whyte, "Quasi-actions on trees II: Finite depth Bass-Serre trees", Memoirs of the American Mathematical Society, 2011, for $n\ge 2$. (The case $n=1$ you already know.) – Moishe Kohan Apr 20 '17 at 22:44