Exercise 5.7.22* (Density of $\langle P \rangle$ if $P$ is on the unbounded convex component)

Proof.

(a)

Suppose that $P\in {𝒞}_0$.

By Problem 21, we know that ${𝒞}_0$ is a subgroup of ${𝒞}_f(ℝ)$. We can do the same reasoning as in Problem 20: the group ${𝒞}_0$ is a compact connected real Lie group of dimension $1$, so is isomorphic to $S_1=ℝ∕\mathbb{Z}$. The subgroups of $S_1$ are finite, or dense on $S_1$. If $P$ has infinite order in ${𝒞}_f(ℝ)$, then $⟨P⟩=\{\mathit{nP},n\in \mathbb{Z}\}\subset {𝒞}_0$ is an infinite subgroup of ${𝒞}_0$, thus its image in $S_1$ is dense on $S_1$. This shows that $⟨P⟩$ is dense on ${𝒞}_0$, because an isomorphism of Lie groups is also an homeomorphism.

(b)

Suppose that $P\in {𝒞}_1$.

By Problem 21, $2P\in {𝒞}_0$, and by induction, for all $k\in \mathbb{N}$, $2\mathit{kP}\in {𝒞}_0$ and $(2k+1)P\in {𝒞}_1$. Therefore, for all $k\in \mathbb{Z}$, $2\mathit{kP}\in {𝒞}_0$ and $(2k+1)P\in {𝒞}_1$.

Consider the subgroup $H=⟨P⟩=\{\mathit{nP},n\in \mathbb{Z}\}$ of ${𝒞}_f(ℝ)$. Then $H\cap {𝒞}_0=⟨2P⟩$ is an infinite subgroup of ${𝒞}_0$. By part (a), $H\cap {𝒞}_0$ is dense on ${𝒞}_0$.

Consider the map

$$\begin{array}{lll}\hfill \eta \{\begin{array}{ccc}\hfill {𝒞}_1\hfill & \hfill \to \hfill & {𝒞}_0\hfill \\ \hfill M\hfill & \hfill \mapsto \hfill & P+M.\hfill \end{array}& & \hfill \end{array}$$

Then $\eta$ is well defined: since $P$ in ${𝒞}_1$, if $M\in {𝒞}_{}$, then $P+M\in {𝒞}_0$ by Problem 21.

Moreover $\eta$ is continuous, because the addition is continuous on ${𝒞}_0$ by the explicit formulas.

$\eta$ is bijective, with reciprocal ${\eta }^{-1}:N\mapsto N-P$. Therefore ${\eta }^{-1}$ is also continuous, so $\eta$ is an homeomorphism.

By our previous remarks, $\eta ({𝒞}_1\cap H)={𝒞}_0\cap H$. Since ${𝒞}_0\cap H=⟨2P⟩$ is dense on ${𝒞}_0$, then ${𝒞}_1\cap H={\eta }^{-1}({𝒞}_0\cap H)$ is dense on ${𝒞}_1$.

Therefore the subgroup $\{\mathit{nP},n\in \mathbb{Z}\}$ is dense on ${𝒞}_f(ℝ)$.