COLLOQUIUM MATHEMATICUM VOL. 125 2011 NO. 2 A FREE GROUP OF PIECEWISE LINEAR TRANSFORMATIONS BY GRZEGORZ TOMKOWICZ (Bytom) Abstract. We prove the following conjecture of J. Mycielski: There exists a free non- abelian group of piecewise linear, orientation and area preserving transformations which acts on the punctured disk {(x, y) ∈ R 2 :0 <x 2 + y 2 < 1} without fixed points. 1. Introduction. In the theory of paradoxical decompositions which goes back to Hausdorff, Banach and Tarski, and von Neumann (see the survey of M. Laczkovich [2] and also [5]) the following conjecture was open until recently: (I) The punctured disk D = {(x, y) ∈ R 2 :0 <x 2 + y 2 < 1} has a paradoxical decomposition relative to the group SL 2 (R). Likewise we have the more demanding conjecture: (II) There exists a partition of D into three sets A,B,C such that the six sets A,B,C,A ∪ B,B ∪ C, C ∪ A are equivalent to each other by finite decomposition relative to the group SL 2 (R). Recall that sets A, B ⊂ X are equivalent by finite decomposition (or equidecomposable ) relative to a group G acting on X if there exist finite partitions {A i } k i=1 and {B i } k i=1 of A and B respectively and g 1 ,...,g k ∈ G such that g i (A i )= B i for each 1 ≤ i ≤ k. The set E ⊂ X is paradoxical relative to G if E contains disjoint subsets A and B and each of them is equidecomposable to E relative to G. Conjecture (I) was proved by M. Laczkovich [1]. (II) presents additional difficulties, and it will be proved in the present paper. In fact it is known (see [4] and Corollary 4.12 in [5]) that with the use of the Axiom of Choice, affirmative answers to (I), (II) and many similar conjectures follow from the following theorem: Theorem 1.1. There exists a free nonabelian group F of permutations acting on the punctured disk D = {(x, y) ∈ R 2 :0 <x 2 + y 2 ≤ r 2 } such that if f ∈ F \{e} and x ∈ D then f (x) = x, and for every f ∈ F there exists a 2010 Mathematics Subject Classification : Primary 03E05, 20E05, 51M05; Secondary 20G20. Key words and phrases : free group, Hausdorff–Banach–Tarski paradox, paradoxical set. DOI: 10.4064/cm125-2-1 [141] c  Instytut Matematyczny PAN, 2011