Torsion-free Lie groups Out of curiosity, I'm looking for some examples of Lie groups that are torsion-free.
For some reason (and perhaps there is a good reason), most torsion-free groups I've heard of are discrete. One Lie example that comes to mind is $\mathbb{R}^n$. Also, if $G$ is a Lie group and its torsion subgroup, call it $\Gamma$, is both normal and discrete, then $G/\Gamma$ provides another source of examples.
Question 1: What are some more examples of torsion-free Lie groups?
Question 2: Are there any examples that have non-trivial compact Lie subgroups?

A connected Lie group that is torsion-free and it is not isomorphic to a group of matrices is the universal cover of $SL_2(\mathbb{R})$. It is essentially the only semisimple (that is, with no infinite connected abelian subgroups) Lie group that is torsion-free. All the other torsion-free connected Lie groups are solvable and diffeomorphic to $\mathbb{R}^n$ for some $n$.
As well-explained above, if a Lie group is torsion-free then its maximal compact subgroup is trivial.
