= Hyperbolic element of PSL2(R)
{title2=$|\operatorname{tr}A|>2$}
= Hyperbolic Möbius transformation
{synonym}
A real <Möbius transformation> represented by $A\in\mathrm{SL}_2(\mathbb R)$ with $|\operatorname{tr}A|>2$. It has two fixed ideal boundary points and translates along the <geodesic> joining them; its <hyperbolic translation length> is $2\operatorname{arcosh}(|\operatorname{tr}A|/2)$. Conjugating by a real <Möbius transformation> puts it in the form $z\mapsto e^\ell z$ with $\ell>0$; its axis is then the imaginary axis.
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