Is the free group of uncountable rank a Lie group?

Question

Does the free group of uncountable rank have a compatible Lie group structure?

Answer 1

Assuming you require your Lie groups to be second countable so you can't just consider it as a discrete group, then no. Indeed, if $G$ is any Lie group of positive dimension, then there exist nontrivial homomorphisms $\mathbb{R}\to G$, given $t\mapsto \exp(tX)$ for any nonzero element $X$ in the Lie algebra of $G$. But if $G$ is a free group, then there are no nontrivial homomorphisms $\mathbb{R}\to G$, since the image of such a homomorphism would be both abelian and free and hence isomorphic to $\mathbb{Z}$, and there are no nontrivial homomorphisms $\mathbb{R}\to\mathbb{Z}$. So a free group cannot be a Lie group of positive dimension.