The Fixed Points and Cross-Ratios of Hyperbolic Möbius Transformations in Bicomplex Space

The hyperbolic Möbius transformations, which have been defined and proved to be hyperbolic conformal in Golberg and Luna-Elizarrarás (Math Methods Appl Sci 2020, https://doi.org/10.1002/mma.7109 ), are generalizations of Möbius transformations in complex space C ( i ) and hyperbolic space D to multidimensional hyperbolic space D n . In this paper, we study the hyperbolic Möbius transformation in bicomplex space B C isomorphic to D 2 in detail, present a conjugacy classification according to the number of fixed points in S L ( 2 , B C ) , and detailedly prove that the cross-ratio is invariant under hyperbolic Möbius transformations. Furthermore, the present paper generalizes the classical results, which have closed relation with fixed points and cross-ratios, to B C and may give new energy for the development of hyperbolic Möbius groups.