Reference request: affine transforms + circle inversion? This problem cropped up in the context of scale-insensitive methods for generating random variables.
Let $X=R^n \cup \{\infty\}$.  Suppose we consider a set of transforms $\cal{T}$ from $X\rightarrow X$.  We construct them by concatenating functions chosen from the following set:


*

*Invertible linear transform ($x \mapsto Ax$, for $A\in GL_n(R)$)

*Translation ($x \mapsto x+b$)

*Circle inversion ($x \mapsto x/|x|^2$; 0 and $\infty$ swap)


The set $\mathcal{T}$ is very similar to the Mobius transformations, which are built from:


*

*Rotation and scaling ($x \mapsto sAx$, for $A\in SO_n(R)$ and $s$ a positive scalar)

*Translation ($x \mapsto x+b$)

*Circle inversion and reflection ($x \mapsto Mx/|x|^2$, where $M$ reflects through the first coordinate; 0 and $\infty$ swap)


I would like to know if $\cal{T}$ has a standard name, and if any of the properties of the Mobius transformations generalize to $\cal{T}$.  For instance, Mobius transformations in $R^2$ preserve generalized circles; are generalized ellipsoids in $R^n$ preserved by $\mathcal{T}$?  Is there a property analogous to the cross-ratio?  Any references would be greatly appreciated.
 A: $\mathcal{T}$ is not a Lie group when $n>1$.  
Actually, the OP did not say whether he wanted $\mathcal{T}$ to be all possible sequences of compositions of these generating sets, but, if he did, then it is clear that $\mathcal{T}$ is not a Lie group, in the sense that it is not defined as the set of solutions of some system of PDE for transformations of $\mathbb{R}^n$.  For one thing, the group that they generate would properly contain the conformal group $\mathrm{O}(n{+}1,1)$ acting on $S^n$, which is known to be a maximal Lie group, i.e., there is no group (in Lie's sense) between the conformal group and the full diffeomorphism group.  (NB: The group of analytic diffeomorphisms of $S^n$ is not a subgroup of the full diffeomorphims in Lie's sense because it is not defined as the set of solutions of some system of PDE.)
In particular, no group $G$ that contains $\mathcal{T}$ can preserve any geometric structures of the kind the OP mentions because this would define a PDE that $G$ satisfies.
(By the way, note that $\mathcal{T}$, as the OP defined it, does not consist of smooth transformations of $S^n$ only when $n>1$, since the non-conformal affine transformations do not extend smoothly to $\infty$ except when $n=1$.)
