We know that the set of order two elements of $R^n$, tori and $S^3$ are discrete. Are there others examples of Lie groups with such property? Are there some characterization of such class?
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2$\mathrm{SL}(2,\mathbb{C})$, $\mathrm{SL}(2,\mathbb{R})$, and any of the above examples $\times $ a finite group. – Oct 01 '14 at 19:00
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1If $G$ is a connected Lie group, then its set of order 2 elements is discrete if and only if every element of order 2 is central. – YCor Oct 01 '14 at 21:12
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A simple characterization is:
If $G$ is a connected Lie group, then its set of order 2 elements is discrete if and only if every element of order 2 is central.
$SU(2)$, $SL_2(\mathbf{R})$, $SL_2(\mathbf{C})$ satisfy this property but not $SO(3)$, $SL_2(\mathbf{R})$, $SL_2(\mathbf{C})$. So it's not an invariant of the Lie algebra, nor of the maximal compact subgroup (since the maximal compact subgroup of $PSL_2(\mathbf{R})$, being abelian, satisfies the property). Also it can't be read on a Levi factor, since $SL_2(\mathbf{R})\ltimes\mathbf{R}^2$ fails to satisfy the property.
YCor
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