Lyn Assy - Basics of Homeo+(S1)

Abstract

(Briefly) discuss examples of projective and piecewise linear homeomorphisms and introduce/recall Thompson’s group. Cover §4 and prove Theorem 4.3 using fragmentation and Mazur swindle following [Mann] pp. 5-6.

Talk

Generally speaking, by studying Algebras acting on spaces we can find out something on dynamics and vice versa.

Example

We look at some spaces acted upon by linear groups. If we want to study dynamics, we need to study groups which do not acts transitively. This is because we are primarily interested in the shape of orbits of a group action on a space. Nice groups are given by the Fuchsian group or the Thompson’s group F.

Definition

We define the Thompson’s Group F be first defining the group of piece-wise linear maps $PL(\mathbb{R})$ on $\mathbb{R}$ and orientation-preserving PL-maps on $\mathbb{R}$. We then define the Thompson’s groups as the group with diadic angles in the graph.

Wait, since this is defined on the circle and not on the interval this is not the Thompson group $F$ but a different one. It differs in the fact that the endpoints do not to be fixed but can be rotated.

We can endow the space of Homeomorphisms with a topology, given by Uniform convergence.

Example

We can define a Thompson group with one repelling fixed point and one contracting fixed point. Note that when the iterate the group, the limit is not a homeo anymore, meaning the space is not closed.

Tip

We state a fact that every Lie group $G$ is topologically homeomorphic to a direct product of a max compact subgroup $K\subseteq G$ and a finitely-dimensional euclidean space. A simple example is Gram-Schmidt-Verfahren: $G{L}_n(\mathbb{R})\cong O(n)\times {\mathbb{R}}^d$

We go back to the homeomorphism space of the circle

Proposition 1

$SO(2,\mathbb{R})$ is a max. compact subgroup of $Home{o}_{+}(S^1)$ (up to conjugacy)

Proof: For the proof, we use the Haar-Maß. This is a volume measure for Lie groups which is invariant under left-multiplication. We take some compact subgroup $K\subseteq G$ We can then do an average-thingy to get a measure on the circle which is invariant under the action of $K$. We state or derive that there are not atoms (i.e. points with positive measure). Somehow the idea of this proof is to show that there is a Lebesgue measure, invariant under $K$. This can only happen, if $K$ is a group of rotations. To get this Lebesgue measure we took the Haar measure and took an average.

Proposition 2

The inclusion $SO(2,\mathbb{R})$ into $Home{o}_{+}(S^1)$ is a Homotopy equivalence.

Proof: We study the group $\tilde{H}=\widetilde{Home{o}_{+}+(S^1)}$ which is a lift of a homeo of $S^1$ to $\mathbb{R}$. We find a retraction $\tilde{r}:\tilde{H}\to \mathbb{R}$. The idea here is to take one element $\tilde{h}$ and homotope it to the rotation element with the same Rotationszahl. This gives us a homotopy of $\tilde{H}$ to $\mathbb{R}$.

Prop 3

The Homeo space is Einfache Gruppe. This means that every homomorphism maps to itself of the trivial group.

Prop 4

For any “generic” pair $f,g$ from the Homeo group the group generted by $f,g$ is Free group.

Proof: For two elements $f,g$ (I image them to be random elements of homeo) we take the word map from a free group. This is given by a word $w\in F=⟨a,b⟩$. This is then map from $H\times H$ to $H$. (the homeos) and given be replacing the two generators $a,b$ of $F_2$ in $w$ by $f,g$. We note that if for each $w\in F_2$ the word map is not the identity, then $f,g$ generate a free group (since there are no relation). We define two subsets $X_w,Y_w$. Where the first set consists of all $(f,g,x)\in H\times H\times S^1$ that fulfill $w(f,g)(x)=(x)$ and the second are all sets where $(f,g)\in H\times H$ where $w(f,g)=id$. The second set detects relations and the first set detects almost relations. Meaning, we have $Y_w\subseteq X_w$ The group $X_w$ is useful as the complement over all $w$ will definitely give us a Free group. We find out that the complement is open and dense, meaning it contains nearly everything and nearly every element generate a free group. We proved this by contradiction.