Reading small actions of a one-ended hyperbolic group on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-trees from its JSJ splitting

In this paper, we show how the canonical JSJ splitting of a one-ended hyperbolic group allows us to understand all its small actions on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-trees: they are obtained by blowing up the surface type vertices into an action corresponding to a measured foliation on the corresponding orbifold; by blowing up elementary type vertices into a finite tree with bounded complexity; and by collapsing some edges. We deduce that every small action of a one-ended hyperbolic group on an [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]-tree is geometric. We also derive a strong uniqueness property of the JSJ splitting.