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

Advertisements

Skip to content
# Free subgroups of the group of homeomorphisms

##
One thought on “Free subgroups of the group of homeomorphisms”

### Leave a Reply

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

Advertisements

%d bloggers like this:

Here is a proof for .

Let be rotations in 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 be a rotation by radians with axis and be a rotation by radians with axis , with irrational). To set notation, say defined is injective.

Extend to the ball: more precisely, pick homeomorphisms of the ball fixing its boundary such that their restriction to the sphere of radius is respectively.

Now the group morphism given by is injective: if , then the restriction of to the sphere of radius is , and so as wanted.

LikeLiked by 1 person