Group G having two ends

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Let $Z$ set of integer is a subgroup of G. $Ends(G)=2$. Is it true that $Z$ is of finite index in G ?
I was trying to show quasi isometry between $Z$ and $G$. For notation and other definition please refer to the book Bridson and Haefliger page 144-146 http://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf