Decomposition Theorems for Triple Spaces

A triple space is a homogeneous space $G\mathrm{/}H$ where $G=G_0\times G_0\times G_0$ is a threefold product group and $H\simeq G_0$ the diagonal subgroup of $G$. This paper concerns the geometry of the triple spaces with $G_0=\text{SL}(2,\mathbb{R})$, $\text{SL}(2,\text{C})$ or ${\text{SO}}_e(n,1)$ for $n\ge 2$. We determine the abelian subgroups $A\subset G$ for which there is a polar decomposition $G=KAH$, and we determine for which minimal parabolic subgroups $P\subset G$, the orbit $PH$ is open in $G\mathrm{/}H$.