What is in the image of the exponential of $\frak{sl}(n,\mathbb{R}$)? What do you need to get all of $\mathrm{SL}(n,\mathbb{R}$)?

Question

This question discusses how $\mathrm{S}L(2,\mathbb{R}$) coincides with $\pm\exp(z)$ with $z\in \frak{sl}(n,\mathbb{R}$) (the real traceless matrices). Is it known what happens for $n>2$?

Namely, one can represent all invertible matrices with determinant $1$ and trace greater than or equal to $-2$. Also, just by providing a sign (more precisely, up to the center of the group, which is $\{ I, -I\}$) one is able to obtain all of $\mathrm{SL}(2,\mathbb{R})$ ).

Is such a result known for the general case of $\mathrm{SL}(n,\mathbb{R})$, which characterizes the image of the exponential and what one needs to "add" in order to obtain the entire group (i.e. what is the "minimal" set $A\subseteq G$ such that $\mathrm{SL}(n,\mathbb{R}) = A \exp(\frak{sl}(n,\mathbb{R}))$ )?

I would also be interested in the more general question regarding classical Lie groups, although this would already be a great help towards a greater understanding of what happens in the exponential (even just for $n=3$).

Answer 1

I am not sure if the complete answer is known, but the interior and the exterior of the image $E$ of the exponential map are known:

1. The interior of $E$ consists of all matrices in $SL(n,\mathbb R)$ which have no negative eigenvalues.

2. The exterior of $E$ consists of all matrices in $A\in SL(n,\mathbb R)$ which have at least one negative eigenvalue of odd multiplicity.

3. Thus, the boundary of $E$ consists of matrices $A\in SL(n,\mathbb R)$ such that some eigenvalues of $A$ are negative but all such eigenvalues have even multiplicity. For instance, $A=Diag(-1/2,-1/2,-2, -2, 1)$ is an example.

This result is due to M.Nishikawa but his paper does not seem to be accessible. Another proof can be found in

Đoković, Dragomir Ž., The interior and the exterior of the image of the exponential map in classical Lie groups, J. Algebra 112, No. 1, 90-109, Corrigendum 115, No. 2, 521 (1988). ZBL0638.22006.

He also gives a description of the image $E$ of the exponential map for $SL(n,\mathbb C)$ but it is a bit too hard to state and you can find it in his paper. The set $E$ is open and dense (for some reason, he does not say "open" but it is open because the exponential map is holomorphic).