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I am currently studying Möbius geometry from the book [1]. I found a group in Möbius geometry called Möbius group which contains Möbius transformations. I have the following doubt.

Dose this group contain transformations of the form $f(z)=\frac{az+b}{cz+d}$ only? or the group contains $f(z)=\frac{1}{\bar{z}}$ type functions also.

Kindly help. It will be great if anyone suggests some references to understand it.

Reference: [1] W. Benz, Classical Geometries in Modern Context, Birkhauser Basel, 2012.

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You seem to imply that the book from which you study does not DEFINE what a Möbius transformation is. Then discard this book and choose another one. Unfortunately the terminology is not settled. The standard definition of Möbius group (in any dimension) is the group generated by inversions in circles. According to this definition, in dimension 2 , $1/\overline{z}$ is a Möbius transformation. The subgroup consisting of orientation-preserving transformations is accordingly called the group of orientation-preserving Möbius transformations. In dimension 2, it consists of linear-fractional transformations.

However many authors restrict the name Möbius transformation to orientation-preserving (conformal) maps, and this creates a confusion.

On my opinion, orientation-preserving Möbius transformations of the plane should be called linear-fractional transformations.

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    $\begingroup$ The book is discussing about Möbius sphere geometry and according to the book, I think $1/\bar{z}$ is in the Möbius group. In some other books, I have seen that the Möbius group for is isomorphic to $O(n+1,1)/\{I,-I\}$ for $n$ dimensional sphere geometry. And in some books authors do not consider the function $1/\bar{z}$. This confused me. $\endgroup$
    – Learning
    Commented Aug 8 at 15:52
  • $\begingroup$ I couldn't agree more with this answer; the implicit restriction by many to the group of oriented Möbius transformations can cause some annoying confusion. $\endgroup$
    – Yousuf Soliman
    Commented Aug 8 at 20:10
  • $\begingroup$ @Learning: I only said that a decent math book has to contain definitions of all terms used in it. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 9 at 2:13

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