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

作者：Chen, Litao; Dai, Binlin^*

来源：Advances in Applied Clifford Algebras, 2022, 32(4): 48.

DOI：10.1007/s00006-022-01231-1

• Bicomplex numbers
• Hyperbolic Mobius transformations
• Fixed point
• Cross-ratio

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