Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality.
My idea: Suppose $X$ satisfies a linear isoperimetric inequality. Let $f : X \rightarrow Y$ be a quasi-isometry with quasi-isometry inverse $g$. Let $c$ be a loop in $Y$. Then $g(c)$ is a loop in $X$. Then using the fact that $X$ satisfies linear isoperimetric inequality we have to show that $Y$ satisfies a linear isoperimetric inequality. But I can't do that.