If $H$ is a subgroup of $G$, then the inclusion is a quasi-isometry if and only if $H$ has finite index in $G$.
I am reading through some notes on Geometric Group Theory and I came across this side fact in the discussion. Any suggestions on how to solve it?
I know a map $f:X \rightarrow Y$ is a quasi-isometry iff it is a quasi-isometric embedding and a quasi-isometric inverse $g:Y\rightarrow X$ such that for all $y \in Y$ there exists $x \in X$ such that $d_X(g\circ f(x),y)< D$ and such that for all $x \in X$ there exists $y \in Y$ such that $d_Y(f\circ g(y),x)< D$. Alternatively $g$ could just be an isometric embedding as well and the definitions coincide.