Free subgroups of the group of homeomorphisms

Let n\geq 2. Prove that the group of homeomorphisms of the n-dimensional ball that fix its boundary contains a copy of the free group on 2 generators.

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Free subgroups of the group of homeomorphisms

One thought on “Free subgroups of the group of homeomorphisms

  1. Here is a proof for n\geq 3.

    Let f,g be rotations in \mathrm{SO}(n) such that they generate a free group (this is a usual construction carried out in the proof of the Banach-Tarski theorem: for instance, let f be a rotation by \theta_1 radians with axis e_1 and g be a rotation by \theta_2 radians with axis e_2, with \theta_1, \theta_2, \theta_1-\theta_2 irrational). To set notation, say \psi: \langle a,b\rangle \to \mathrm{SO}(n) defined a\mapsto f, b\mapsto g is injective.

    Extend f,g to the ball: more precisely, pick F,G homeomorphisms of the ball fixing its boundary such that their restriction to the sphere of radius 1/2 is f,g respectively.

    Now the group morphism \varphi:\langle a,b\rangle \to \mathrm{Homeo}(D^n, \partial D^n) given by a\mapsto F, b\mapsto G is injective: if w\in\ker\varphi, then the restriction of \mathrm{id}=\varphi(w) to the sphere of radius 1/2 is \psi(w), and so w=1 as wanted.

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