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.
2026-03-28 14:06:32.1774706792
Do quasi-isometries of semi direct products of free groups preserve fibres
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