Polish Group Topologies on PSL(2,C)

Question

Does anyone know how many Polish group topologies (or where to begin to look for this information) can be put on $\text{PSL}_2(\mathbb C)$?

Answer 1

Let me make a partial answer. I am working in ZFC until an inconsistency will be found.

Any non-continuous automorphism of the field $\mathbb{C}$ gives a non-continuous automorphism of $G=\text{PSL}_2(\mathbb{C})$. Pulling back the standard topology by such will provide a new locally compact second countable (lscs) group topology on $G$. In fact, you may consider doing the same with an arbitrary group automorphism of $G$, but it can be shown that any such group automorphism is actually given by a field automorphism. This fact is proved by Borel-Tits in "Homomorphismes “abstraits” de groupes alg´ebriques simples" and also in a more recent paper by Linus Kramer which is very relevant for your question http://arxiv.org/pdf/1009.5457v6.pdf, see lemma 16 (note that most of the paper deals with the "absolutely simple" situation, a property $G$ fails to have). There are tons of autos of $\mathbb{C}$, so if what you're after is cardinallity, you'd be satisfied with that.

All these topologies are lcsc, in fact Lie. There are more. There are more Polish field topologies on $\mathbb{C}$, for example by identifying it with $\mathbb{C}_p$, and any such will give you a new topology on $G$ (which will be Polish, but not lcsc). Regarding these, see my question Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value? At present time, I got no answer... I am very interested in knowing all these topologies. Hey, reader: go there and write your answers!

I suspect the "list" above is full, all topologies are coming from field topologies, but I don't see an argument for this currently. Note that you can emebed the additive group $(\mathbb{C},+)$ at the upper-right corner and get a new group topology on it by introducing one on $G$. In the case of $\text{PGL}_n$, $n\geq 3$ it is not hard to see that you'd get a field topology on $\mathbb{C}$, but for $n=2$ I am not sure.

Let me add one more relevant remark. $G$ with the standard topology has the property that for every continuous homomorphism to every topological group $H$ the image is closed. This is an old theorem of Omori. It follows that if your new topology is known to be weaker than the standard one, then they must coincide. The latter is generally correct for Polish topology, but in this case it applies to ANY new topology.