# Expressing every element of a lie group as a product of exponentials?

Given a connected lie group $$G$$, since a neighbourhood of the origin generates all of $$G$$, we have that every $$g$$ in $$G$$ can be expressed as a finite product of elements of the form $$e^X$$, for $$X$$ in the lie algebra $$\mathfrak{g}$$ of $$G$$.

Let us define $$f(G)$$ to be the minimal $$n\in \mathbb{N}\cup \{\infty\}$$ such that any element of $$G$$ can be expressed as a product of $$n$$ elements of the form $$e^X$$ for $$X$$ in $$\mathfrak{g}$$.

It's a theorem that for compact $$G$$, we have $$f(G)=1$$, which is another way of saying $$exp$$ is surjective, and we can check that $$exp$$ is surjective for the the group of upper(or lower) triangular matrices with $$1$$s on the diagonal, so by factoring any matrix in $$GL_n(\mathbb{C})$$ as $$UDU'$$ with $$U,U'$$ of this form, and $$D$$ diagonal, we also see that $$f(GL_n(\mathbb{C}))\leq 3$$.

One can do similar things with $$SL_n$$, and the group of upper triangular matrices, and get that in each case, $$f(G)$$ is finite.

Is $$f(G)$$ always finite? What about for linear lie groups?