Abstract
In this work we study and provide a full description, up to a finite index subgroup, of the dynamics of solvable complex Kleinian subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \). These groups have simple dynamics, contrary to strongly irreducible groups. Because of this, we propose to define elementary subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \) as solvable groups. We show that triangular groups can be decomposed in four layers, via the semi-direct product of four types of elements, with parabolic elements in the inner most layers and loxodromic elements in the outer layers. It is also shown that solvable groups, up to a finite index subgroup, act properly and discontinuously on the complement of either a line, two lines, a line and a point outside of the line, or a pencil of lines passing through a point. These results are another step towards the completion of the study of elementary subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \).
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Acknowledgements
The author would like to thank to the IMUNAM Cuernavaca, CIMAT, FAMAT UADY and their people for their hospitality and kindness during the writing of this paper. Finally, the author is specially thankful to Ángel Cano, Mónica Moreno, Waldemar Barrera, Juan Pablo Navarrete, Carlos Cabrera and Manuel Cruz for many valuable and helpful conversations.
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Toledo-Acosta, M. The Dynamics of Solvable Subgroups of \(\text {PSL}\left( 3,\mathbb {C}\right) \). Bull Braz Math Soc, New Series 53, 127–171 (2022). https://doi.org/10.1007/s00574-021-00254-9
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DOI: https://doi.org/10.1007/s00574-021-00254-9