$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$Consider the group $\GL_+(3,\mathbb{R})$ acting on $\mathbb{R}^3$. It induces an action of $\PGL_+(3,\mathbb{R})$ on the "sphere at infinity". Moreover, the group of Moebius transformations $\PSL(2,\mathbb{C})$ of the sphere at infinity is a $6$-dimensional subgroup of $\PGL_+(3,\mathbb{R})$. Thus the $\PGL_+(3,\mathbb{R})$-orbit of the standard conformal structure $c$ on the $2$-sphere at infinity is $2$-dimensional, and can be described as $$ \PGL_+(3,\mathbb{R}) / \PSL(2,\mathbb{C}). $$

Could someone perhaps describe this homogeneous space more concretely? As a wild guess, is it perhaps diffeomorphic to $S^2$ itself?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$Consider the group $\GL_+(3,\mathbb{R})$ acting on $\mathbb{R}^3$. It induces an action of $\GL_+(3,\mathbb{R})/\mathbb{R}_+$, which is isomorphic to $SL(3,\mathbb{R})$, on the "sphere at infinity". Moreover, the group of Moebius transformations $\PSL(2,\mathbb{C})$ of the sphere at infinity is a $6$-dimensional group (over $\mathbb{R}$). What is the $\SL(3,\mathbb{R})$-orbit of the standard conformal structure $c$ on the $2$-sphere at infinity?

Could someone perhaps describe this homogeneous space more concretely? (I took LSpice's and IanAgol's comments into consideration and re-edited my post!).