TITLE: universal cover of SL2(R): does it admit central extensions?
QUESTION [17 upvotes]: Is it true that the universal cover of SL2(ℝ) has no non-trivial central extensions... as an abstract group?
(that's certainly true as a Lie group)
Motivation:
I have a projective action of SL2(ℝ) on some Hilbert space H
and I'd like to know that it induces an honest action of its universal cover.
But it's a hassle to show that the action is continuous.
So I'm wondering if there is an alternative argument that uses solely group theory.

REPLY [10 votes]: As van der Kallen mentions, the answer is: definitely no. Here's argument which is less precise than his, but with a checkable basic reference: thm 11.10 from Milnor's book "Introduction to Algebraic K-theory": if a field F is uncountable then $K_2(F)$ is uncountable as well. Moreover $K_2(F)$ is generated by symbols which already define cocycles on $\text{SL}_2$. So the Schur multiplier of the discrete group $\text{SL}_2(\mathbf{R})$ is uncountable, hence certainly not cyclic.
In the same spirit, the Schur multiplier of the simply connected Lie group, $\text{SL}_2(\mathbf{C})$, viewed as a discrete group, is uncountable.