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A free group of piecewise linear transformations

Grzegorz Tomkowicz — 2011

Colloquium Mathematicae

We prove the following conjecture of J. Mycielski: There exists a free nonabelian group of piecewise linear, orientation and area preserving transformations which acts on the punctured disk {(x,y) ∈ ℝ²: 0 < x² + y² < 1} without fixed points.

The Banach-Tarski paradox for the hyperbolic plane (II)

Jan Mycielski; Grzegorz Tomkowicz — 2013

Fundamenta Mathematicae

The second author found a gap in the proof of the main theorem in [J. Mycielski, Fund. Math. 132 (1989), 143-149]. Here we fill that gap and add some remarks about the geometry of the hyperbolic plane ℍ².

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