Commands
Abstract
State the general Tits alternative of Margulis for as given in [Ghys] §5.2 and indicate the steps involved in the proof. Discuss the main results of Farb-Shalen stated in the introduction of [FS]. Mention the example of a -action of the punctured mapping class group ([FS] §6.2) which cannot be improved to be analytic. Prove Theorem 1.1, without proving Holder’s Theorem, as this will be proved in a later talk. Prove Theorem 1.3 and deduce Theorem 1.4. Note: Here one can also swap the presentation to cover the analytic case first.
Talk
We explain what virtually and residually means. Virtually means a finite index subgroup has a property. Residually means many quotients of the group have the property. We also explain what a Auflösbare Gruppe is.
The main theorem is Tits Alternative:
Theorem
Take a subgroup from a general linear space. Then exactly one of the two options happens:
- contains a non-abelian free subgroup
- is virtually solvable
There is a chain of subgroup
\text{Diff}_{+}^{\infty}(S^{1}) \subset \text{Homeo}_{+}(S^{1})$$ So does this apply to higher groups? Yes, there is an analogous statement! >[!tip] Theorem (Residual Tits alternative) > >Let $G$ be a subgroup of $Diff_{+}^{w}(S^{1})$, i.e. the analytic functions. Then either (**not exclusive**) >1. $G$ contains a non-abelian free group >2. $G$ is residually solvable. In fact there is a surjective map $\phi: G \to C_{n}$ so that $[ker \phi, ker \phi]$ is residually torsion-free nilpotent. >[!tip] Corollary > >Any non-trivial perfect subgroup of $Diff_{+}^{w}(S^{1})$ has a nonabelian free quotient. >This apparently means $[G, G] = G$. The [[Thompson's group F]] has been mentioned as an interesting counter-example or example. >[!tip] Example > >We study the mapping class group. Lifting the surface to the [[Hyperbolische Halbebene]] (and using one marked point to fix something we can apply mapping classes and those act on the circle. For Genus $\geq 4$ and punctures $\geq 0$ the mapping class group maps trivially into $\D$. **Proof of the residual alternative:** If $G$ is a subgroup of $\D$ that fixes some point $x$ on the circle. Then $[G, G]$ is residually (torsion free)-nilpotent group. We study the action of $G$ on the [[k-Jet]] at $x$. (Jets are a weird generalisation of tangent vectors, retaining higher derivatives). >[!tip] Lemma > >Let $G$ be a subgroup of $\D$. Then this action preserves the ordering of a finite set of points and therefore acts as a cyclic group. This means $G$ factors through $C_{n}$. (Assuming there is a finite orbit) >[!tip] Theorem > >If $G$ acts freely on $\mathbb{R}$ then it is [[Abelian group]]. This follows from [[Holder's Theorem]]. >[!tip] Proposition > >We quickly set the [[Coset lemma]] which tells that if we have a finite union of cosets (possibly of different subgroups) then one of the subgroups must be finite index. (Its slightly more complicated) **Proof of the Tits alternative (for real!):** We take $G$ a subgroup of $\D$. If $G$ acts freely on the circle then $G$ is abelian. (maybe because there arent any fixed points, meaning the dynamics is conjugate to a rotation, assuming there are dense orbits). If not, we can find an element in $G$ with finitely many fixed points (due to analytic properties, they are finite and discrete). We look at the segments sitting between the fixed points. The segments are shifted (without fixed points) giving us a kind of repelling and attracting action. We can show this by defining small segments at the boundary and repeat the action often enough to move the side segments into another. Taking a second action with disjoint fixed point sets, the [[Ping-Pong Lemma]] then tells us that there are two hyperbolic elements.