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

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

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.

• Thanks for saving me that last minute of typing (+1). Discarding 3 lines is fine :-) Commented Oct 27, 2015 at 7:47
• Just curious, is there any compact example?
– user99914
Commented Oct 27, 2015 at 7:54
• $SO(3)$ is $\mathbb S^3 /\{\pm 1\}$, so it's a double cover of a Lens space $\mathbb S^3/\mathbb Z_4$ so it seems that it has such a homeomorphism (I do not know the answer too).
– user99914
Commented Oct 27, 2015 at 8:29
• @JohnMa $SO(3)$ has an element of order 2 so there is a free action by $Z/2Z$ Commented Oct 27, 2015 at 9:02
• Every compact Lie groups must have a toric subgroup $\mathbb S^1$, that $-1$ in $\mathbb S^1$ will be an order two element. @AliTaghavi
– user99914
Commented Oct 27, 2015 at 9:13