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I am looking for information about the group $G$ generated by Mobius transformations and complex conjugation, $z \to\bar{z}$. In other words, transformations of the form either $z \to \frac{az+b}{cz+d}$ or $z \to \frac{a\bar{z}+b}{c\bar{z}+d}$ for $a,b,c,d\in \mathbb{C}$ which represent an element of $PGL(2,\mathbb{C})$, the Mobius group.

Note that inversions in circles are included in this group. In particular, the formula for inversion in a circle at point q of radius R is given by $z \to \frac{q \bar{z} + (R^2-|q|^2)}{\bar{z}-\bar{q}}$ on page 125 of Visual Complex Analysis by Needham. This formula is the composition of a Mobius transformation and a complex conjugation.

This group $G$ is a $\mathbb{Z}_2$ extension of the Mobius group. The Mobius group is a normal subgroup of $G$, because if $m$ is a mobius transformation, then whether $g \in G$ is orientation-preserving or reversing, $gmg^{-1}$ is orientation-preserving, so $gmg^{-1} \in PGL(2, \mathbb{C})$. We have:

$0 \to PGL(2,\mathbb{C}) \to G \to \mathbb{Z}_2 \to 0 $

Question 1) Is there a nice matrix representation of the group $G$? Does it have a standard name in the literature?

Question 2) This group preserves all the lines and circles in the plane. Is $G$ the full group that preserves lines and circles or are there other more exotic transformations?

References: This question is about circle inversions, but doesn't address the group theory. Explicitly representing an isometry as a composition of circle inversions

There are a number of references online that refer to "the inversive group" but I couldn't find any detailed information.

Daniel Longenecker
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These are frequently also called Moebius transformations. The group they form is called the general Moebius group. It is naturally isomorphic to the full isometry group of the 3-dimensional hyperbolic space. You can find its detailed discussion (including higher-dimensional case) for instance in

Beardon, Alan F., The geometry of discrete groups, Graduate Texts in Mathematics, 91. New York - Heidelberg - Berlin: Springer-Verlag. XII, 337 p. DM 108.00; $ 44.60 (1983). ZBL0528.30001.

Ratcliffe, John G., Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149. Cham: Springer. 800 p. (2019). ZBL1430.51002.

Every bijection of the sphere which send circles to circles belongs to this group (the same in higher dimensions). See for instance

Jeffers, Jason, Lost theorems of geometry, Am. Math. Mon. 107, No. 9, 800-812 (2000). ZBL0984.51005.

or page 91 in

Coxeter, H. S. M., Introduction to geometry. 2nd ed, New York etc.: John Wiley and Sons, Inc. XVI, 469 p. (1969). ZBL0181.48101.

This group has faithful matrix representation as the index 2 subgroup of the Lorentz group $O(3,1)$ preserving each component of the 2-sheeted hyperboloid. (Ditto in higher dimensions.)

Moishe Kohan
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