# Anderson's Trick

October 30, 2022

This week I had the pleasure of attending a seminar talk Nick Vlamis gave at CUNY, where he taught us a very pretty trick due to Anderson which one can use, as Anderson did, to prove that the groups of orientation-preserving homeomorphisms of the $2$ - and $3$ -spheres are simple. The purpose of this post is to reproduce Nick’s exposition of the trick.

Suppose $g$ is a homeomorphism of a surface $S$ (for me a surface is a Hausdorff and second countable $2$ -manifold) whose support

$\operatorname{Supp}(g) = \overline{\{x \in S : g(x) \ne x\}}$

is contained in the interior of some closed disk $D$ in $S$ . Since $\operatorname{Supp}(g)$ is closed, there is in fact some closed disk $\Delta$ properly contained in $D$ and containing $\operatorname{Supp}(g)$ . In fact, we can find a countable sequence of disjoint disks $\{\Delta_n\}_{n \in \mathbb{Z}}$ with $\Delta = \Delta_0$ , each contained in the interior of $D$ . Since a sequence of points, one from each disk must have a limit point in $D$ , why don’t we arrange the $\Delta_n$ to Hausdorff converge to a point $x_{+\infty}$ as $n \to +\infty$ and a point $x_{-\infty}$ as $n \to -\infty$ . Here’s the picture. Here’s how we’ll use these disks. There is a homeomorphism $\varphi\colon D \to D$ which shifts each $\Delta_n$ to $\Delta_{n+1}$ and holds $x_{\pm\infty}$ fixed. Since the $\Delta_{n}$ are pairwise disjoint, if we choose distinct $n$ and $m$ , the homeomorphisms $\varphi^n g \varphi^{-n}$ and $\varphi^m g \varphi^{-m}$ have disjoint support and thus commute. Thus the infinite product $\sigma = \prod_{n = 0}^\infty \varphi^n g\varphi^{-n}$ makes sense as a homeomorphism whose support is contained in $D$ .

Proposition.

1. $[\sigma,\varphi] = \sigma\varphi\sigma^{-1}\varphi^{-1} = g$ .
2. Suppose $f$ is a homeomorphism of $S$ satisfying $f(D) \cap D = \varnothing$ and such that there exists a homeomorphism $\psi\colon S \to S$ exchanging $D$ and $f(D)$ and satisfying $\psi|_D = f$ and $\psi|_{f(D)} = \varphi f^{-1}$ . Then any normal subgroup of $\operatorname{Homeo}(S)$ containing $f$ contains $g$ .