# Non-surjectivity of $\exp$ map for non-compact Lie group.

I have often read that the exponential map from the Lie algebra of a non-compact Lie group is not surjective, however the product of exponentials involving the compact and non-compact generators of the algebra is surjective. At the moment such a proof is beyond my capability, might someone be able to give me a reference to this result (preferably the original paper, I think it may have been due to Cartan)?

For some non-compact Lie groups such as $\text{SL}_2(\Bbb R)$ it is the case that not every group element is a power of a Lie group element. See Proving that any element of $\text{SL}(2,\mathbb{R})$ can be expressed as $\pm\exp(z)$.
On the other hand, there are non-compact groups such as $\Bbb R^n$ (under addition) where every group element is a power of a Lie group element.
• Hi Lord Shark, that thread you linked was posted from my other account. Are you saying that non all elements of a non-compact lie group can be represented as a product of exponentials from the Lie algebra? The link below claims the contrary, that all elements of $\text{SL}(2,\mathbb{R})$ can be represented as a product of two exponentials. One involving the compact generator and one involving the non-compact generators of $\mathfrak{sl}(2,\mathbb{R})$. Although I haven't been able to find a reference for this claim... physics.drexel.edu/~bob/LieGroups/LG_07.pdf – SigmaAlpha Oct 19 '17 at 6:50
• I think $\text{SL}(2,\mathbb{R})$ is a special case... since we can express any $N\in\text{SL}(2,\mathbb{R})$ as $\pm\exp(z)$ for some $z\in\mathfrak{sl}(2,\mathbb{R})$ and since $-Id$ is in the image of the exponential, it follows that any element can be expressed as the product of two exponentials.. right ? But I am looking for something much more general than this, something about non-compact groups in entirety. – SigmaAlpha Oct 19 '17 at 6:55
• @SigmaAlpha That's not so special. In fact, every element of every real connected Lie group with Lie algebra $\mathfrak g$ can be written as $\exp(X).\exp(Y)$, for some $X,Y\in\mathfrak g$. This was proved in 2003 by Martin Moskowitz and Richard Sacksteder. – José Carlos Santos Oct 19 '17 at 12:48