# Lie groups locally isomorphic to $\operatorname{SL}_2(\mathbb{R})$

I was reading "Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces." They prove a certain theorem for Lie groups locally isomorphic to $$\operatorname{SL}_2(\mathbb{R})$$ which are connected and have finite center.

But they only consider three cases: $$\operatorname{SL}_2(\mathbb{R}),$$ $$\operatorname{PSL}_2(\mathbb{R}),$$ and finite covers of $$\operatorname{PSL}_2(\mathbb{R}).$$ Are these cases exhaustive of all Lie groups locally isomorphic to $$\operatorname{SL}_2(\mathbb{R})$$?

• There are also disconnected groups, as well as the universal cover of $SL(2,R)$. Jul 30 at 23:17
• @MoisheKohan Ah sorry, I forgot to say (just added) that I'm assuming my group is connected w/ finite center.
– user147556
Jul 30 at 23:20
• Then there is nothing left, of course, since $\pi_1(PSL(2, {\mathbb R}))\cong {\mathbb Z}$ and the universal cover has infinite center. Jul 30 at 23:49
• See for instance here. Jul 31 at 0:07

Here are some basic facts about Lie groups.

1. Every connected Lie group $$G$$ has a unique (up to an isomorphism) universal covering $$\pi: \tilde{G}\to G,$$ which is both a covering map and a Lie group homomorphism, necessarily with discrete kernel $$\Lambda$$.

2. The subgroup $$\Lambda$$ is central in $$\tilde{G}$$. Indeed, for each $$h\in \Lambda, g\in \tilde G$$, the commutator $$c_h(g)=[g,h]$$ lies in the kernel of $$\pi$$, i.e. in the discrete subgroup $$\Lambda< \tilde G$$. By continuity and connectivity of $$\tilde{G}$$, the map $$c_h$$ is constant. Since $$c_h(e)=e$$, it follows that $$c_h(g)=e$$ for all $$g\in \tilde G$$, hence, $$h$$ is central in $$\tilde G$$.

3. If $$G_1, G_2$$ are two locally isomorphic connected Lie groups (equivalently, groups with isomorphic Lie algebras), then their universal covering groups are isomorphic. This is the only nontrivial fact I will be using, one of Lie's theorems.

4. From 1 and 2 it follows that for every connected Lie group $$H$$ locally isomorphic to $$G$$, there exists a discrete subgroup $$\Lambda_H$$ of the center $$Z_{\tilde G}$$ of $$\tilde G$$ such that $$H\cong \tilde G/\Lambda_H$$.

5. Suppose now that the center of $$\tilde G$$ is a discrete subgroup $$\Lambda$$, equivalently, the Lie algebra of $$G$$ has trivial center. For instance, the universal covering group of $$SL(2, {\mathbb R})$$ has this property. (Since $$sl(2, {\mathbb R})$$ is centerless.)

Then every connected Lie group $$H$$ locally isomorphic to $$G$$ is isomorphic to a quotient $$\tilde G/\Lambda_H$$ for some subgroup $$\Lambda_H< \Lambda$$. The quotient $$\tilde G/\Lambda$$ is the unique connected Lie group locally isomorphic to $$\tilde G$$ which has trivial center. For instance, in the case of the Lie algebra $$sl(2, {\mathbb R})$$ this smallest quotient will be the adjoint group of $$SL(2, {\mathbb R})$$, namely, $$PSL(2, {\mathbb R})$$.

1. Now, to the specific case of connected groups $$G$$ with the Lie algebra $$sl(2, {\mathbb R})$$. All these groups appear as covering groups of $$PSL(2, {\mathbb R})$$. Since $$\pi_1(PSL(2, {\mathbb R})\cong {\mathbb Z}$$, every such $$G$$ will be either a finite cyclic covering group of $$PSL(2, {\mathbb R})$$ or the unique infinite cyclic covering, the universal covering $$\widetilde{SL}(2, {\mathbb R})$$.

The conclusion is that in the book you are reading, they exhausted the list of all the connected groups with finite center locally isomorphic to $$SL(2, {\mathbb R})$$.