Outer Automorphisms of PSL2(R)

Question

As far as I've been able to tell, a description of Out$(PSL_2(\mathbb{R}))$ isn't available online. I also looked in Lang's $SL_2(\mathbb{R})$ but it's not discussed. I guess my first question would be whether anybody knows of a reference, but if not, does anybody know of an example of at least one explicit element contained in it? Or possibly a subgroup that it must contain, e.g. some free group.

I also couldn't find references for Out$(SL_2(\mathbb{R}))$ or Out$(PSL_2(\mathbb{Q}))$. Does anyone know a book that has some such results?

Thanks a lot!

Answer 1

For one explicit element you can consider the following :

$$\psi :PSL_2(\mathbb{R})\rightarrow PSL_2(\mathbb{R})$$

$$[A]\mapsto [(A^{-1})^t] $$

i.e. you take the inverse and then the transpose (or vice-versa since both operations commute). Since each operation is an anti-morphism, the composition of both will be a morphism. It is an involution so it is an automorphism.

Finally, even if for each $[A]$ we have that $A$ is conjugate with $\psi([A])$, it is easily seen that $\psi$ cannot be inner (by contradiction, if $\psi$ is the inner automorphism by $[A_0]$ look at $[A_0AA_0^{-1}]$ when $A$ is diagonal and when $A\in PSO_2(\mathbb{R})$). `

Edit : warning. Considering Derek Holt's comment. The corresponding automorphism of $SL_2(\mathbb{R})$ is given by the conjugation by an element of determinant -1. Try : $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$.

Another way to construct outer automorphisms is to consider automorphisms of the field. Let us be more abstract to see what I mean by this. Let $L/K$ be a field extension, denote by $G$ the Galois group of $L$ over $K$. To any $\sigma\in G$ one can associate :

$$\psi_{\sigma}:SL_n(L)\rightarrow SL_n(L) $$

$$A\rightarrow \sigma.A$$

Where $\sigma.A$ denotes the matrix whose $(i,j)$-th coefficient is given by $\sigma(A_{i,j})$. It is clear that $\sigma\mapsto \psi_{\sigma}$ is faithfull. Furthermore, if $\psi_{\sigma}$ is inner, i.e. conjugation by $A_0$ then $A_0$ must be in the centralizer of $SL_n(K)$ which has every reason te be trivial so $\psi_{\sigma}$ cannot be inner.

As a matter of fact, this last one is totally irelevant in the present case since $Gal(\mathbb{R}/\mathbb{Q})$ is trivial. However, I think it is (somehow) related to your question so I wanted to add this (think about $Out(SL_2(\mathbb{C}))$, it will contain $Gal(\mathbb{C}/\mathbb{Q})$ which projects on the absolute Galois group).

I would personnally be inclined to think that within "reasonable" hypothesis those are the only way to generate outer automorphisms for $SL_n$, but I have got nothing to back up.

Edit : A reference. "on the automorphisms of the classical groups" Dieudonné, Hua. Memoirs of the AMS number 2 (1951).