On the center of the group of quasi-isometries of the real line

Let $QI(\mathbb{R})$ denote the group of all quasi-isometries $f:\mathbb{R}\to \mathbb{R}\mathrm{.}$ Let $Q_{+}(\text{and }{Q}_{-})$ denote the subgroup of $QI(\mathbb{R})$ consisting of elements which are identity near $-\mathrm{\infty }$ (resp. $+\mathrm{\infty }$). We denote by $Q{I}^{+}(\mathbb{R})$ the index $2$ subgroup of $QI(\mathbb{R})$ that fixes the ends $+\mathrm{\infty },-\mathrm{\infty }$. We show that $Q{I}^{+}(\mathbb{R})\cong Q_{+}\times Q_{-}$. Using this we show that the center of the group $QI(\mathbb{R})$ is trivial.