Lie groups with no free $\mathbb{Z}/2\mathbb{Z}$ action

Question

What is an example of a Lie group which does not have a fixed point- free homeomorphism of order 2?

Answer 1

The group $\mathbb{R}$ works. To see that, note that any homeomorphism $\mathbb{R} \to \mathbb{R}$ of order two must be decreasing, so its graph intersects the line $y = x$, so $f$ has a fixed point.

As pointed out by John Ma in the comments, we cannot take the Lie group to be compact, since any compact Lie group contains a non-trivial torus, and therefore an element of order two. Multiplication by that element then gives a homeomorphism of order two.