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We classify, up to conjugation, all subgroups of the semidirect products ${\mathbb{H}}_1⋊SL\left(2,\mathbb{R}\right)$ and ${\mathbb{R}}^2⋊SL\left(2,\mathbb{R}\right)$. Our methods can also be applied to all Lie groups that are locally isomorphic to them.

The action of the conformal group $O\left(1,n+1\right)$ on ${\mathbb{R}}^n\cup \left\{\mathrm{\infty }\right\}$ may be characterized in differential geometric terms, even locally: a theorem of Liouville states that a $C^4$ map between domains $U$ and $V$ in ${\mathbb{R}}^n$ whose differential is a (variable) multiple of a (variable) isometry at each point of $U$ is the restriction to $U$ of a transformation $x\to g\cdot x$, for some $g$ in $O\left(1,n+1\right)$. In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group $G$ on the space $G/P$ , where $P$ is a parabolic subgroup. We solve...