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I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive homotheties of $\mathbb{R}^{n-1}$. It is well-known that some left-invariant Riemannian metric on it is isometric to the $n$-dimensional hyperbolic space, i.e., has constant curvature $-1$.

What is an example of a Lie group $G$ not isomorphic to the Poincaré $n$-half-space group for any $n$, but satisfying the following:

For every left invariant metric $g$ on $G$, there exists a conformal rescaling $h$ of $g$ such that at each point $x\in G$ all $g$-geodesic rays initiating from $x$ have finite $h$-length, except exactly a single one.

The latter property is satisfied in the upper half plane, the conformal rescaling being the Euclidean metric of $\mathbb{R}^n$, and the exceptional direction being (at any point) the positive vertical one.

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    $\begingroup$ the Poincare upper half space is not a LIe group in a natural sense. $\endgroup$
    – Venkataramana
    Commented Aug 18, 2019 at 1:47
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    $\begingroup$ Upper half-spaces' geometries are not the geometry coming from the semi-direct product of $\mathbb R^n$ and $\mathbb R^+$. That semi-direct product is just a parabolic subgroup of the isometry group $O(n,1)$, which determines the geometry. Then the upper-half space is an $O(n,1)$-space on which $O(n,1)$ acts transitively, isomorphic to the quotient $O(n,1)/(O(n)\times O(1))$. $\endgroup$
    – paul garrett
    Commented Aug 18, 2019 at 3:20
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    $\begingroup$ @paulgarrett I don't really agree... the upper half space is precisely a model of hyperbolic geometry for which there is a canonical choice of point at infinity. It sounds like saying that the real $\mathbf{R}$ line has no zero, because the real line is the affine line, which has a transitive isometry group etc. The upper half plane is perfectly suited to describe the semidirect product described by the OP. $\endgroup$
    – YCor
    Commented Aug 18, 2019 at 4:17
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    $\begingroup$ @YCor, I can understand that there could be reasons to take that viewpoint, but for my purposes, anyway, it is far more useful to see that a physical space is $G/K$, rather than (via an Iwasawa decomposition) $P/(P\cap K)$. Yes, the fact that parabolics have some semi-direct product structure is very useful, and I do use "Iwasawa coordinates" on $G/K$, but I don't ignore $G$ at all. Probably other people have other goals, etc. $\endgroup$
    – paul garrett
    Commented Aug 18, 2019 at 5:04
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    $\begingroup$ The natural generalization of your Poincaré half-space groups would be the well.studied notion of Heintze groups, which are certain sem-direct groups of nilpotent groups (horospheres) with R and which are known to be exactly the homogeneous spaces of negative curvature. Examples include the noncompact noneuclidean symmetric spaces of rank 1. You may try to generalize computations from hyperbolic plane to this setting, replacing the horosphere by a more general nilpotent group. $\endgroup$
    – ThiKu
    Commented Aug 18, 2019 at 9:07

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