# 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.

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.