# Is there a group between SO(2,R)SO(2,\mathbb{R}) and SL(2,R)SL(2,\mathbb{R})?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ?

My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group between, but I can’t point out one.

Note : in the complex case, $H:= SO(2,\mathbb{C}) \cup \left\{ \left( \matrix{ a&b \\ b&-a} \right) ~|~ a^2+b^2=-1 \right\}$ is an example.

I think there’s a fairly simple bare hands proof, even when you don’t require $H$ to be a Lie group.
Consider the image of the unit circle under an element of $SL(2,\mathbb{R})$. This is always an ellipse, centred on the origin, where the product of the lengths of the major axis and minor axis is equal to $4$. The elements of $SO(2,\mathbb{R})$ are precisely those giving a circle (i.e., a major axis of length $2$).
If $H$ contains $SO(2,\mathbb{R})$ and also another element $A$ of $SL(2,\mathbb{R})$ which gives an ellipse with major axis $2a$, where $a>1$, then by pre-composing and post-composing $A$ with suitable rotations, we get all elements of $SL(2,\mathbb{R})$ giving an ellipse with major axis $2a$. So we can assume $A=\begin{pmatrix}a&0\\0&\frac{1}{a}\end{pmatrix}$.
Then $H$ also contains $AR_\theta A$ for every $\theta$, where $R_\theta$ is a rotation through an angle of $\theta$. For $\theta=0$ this element maps the unit circle to an ellipse with major axis $2a^2$, and for $\theta=\frac{\pi}{2}$ to a circle. By the Intermediate Value Theorem, for intermediate values of $\theta$ we can get any major axis length between $2$ and $2a^2$, and by the argument above, $H$ must then contain all elements of $SL(2,\mathbb{R})$ that give an ellipse with major axis between $2$ and $2a^2$.