A whole new definition of geometry was given by Felix Klein when he developed his Erlangen program in the year 1872, under which geometry was described as those properties that remain invariant or simply do not change under a group of transformations. Following the Erlangen program, we have studied the SL(2; ℝ)-action on the space of dual numbers through Möbius transformation. We have classified SL(2; ℝ) into three subgroups through the Iwasawa decomposition that defined three distinct actions on dual numbers. We have found various SL(2; ℝ)-invariant properties of this geometry related to the stabilizer of the dual unit. The concepts of cycles and Fillmore-Springer-Cnops construction have been discussed to show invariant properties of the stabilizer. Lastly, we have discussed the projective cross-ratio of dual numbers and its various invariant and cyclic properties analogous to that of cross-ratio in case of complex numbers.

Topics
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