= Composition of reflections in ultraparallel hyperbolic lines
The composition of <hyperbolic reflections> in two <ultraparallel hyperbolic lines> has infinite order. Map their common perpendicular to the <imaginary axis> in the <Poincaré half-plane model>. The two reflecting <hyperbolic lines> become semicircles centred at zero, with distinct radii $r_1,r_2$. Reflection in the radius-$r$ semicircle is $z\mapsto r^2/\overline z$, so their composition is $z\mapsto(r_2/r_1)^2z$. Every positive power remains nonidentity, because $(r_2/r_1)^{2k}\ne1$ for $k\ge1$.
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