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On free group generated by two Heisenberg translations

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Let A and B be two Heisenberg translations of \(\mathrm{PU}(2,1)\). In this paper, we will discuss the groups \(\langle A, B \rangle \) generated by two non-commuting Heisenberg translations and determine when they are free. The main result of the paper improves an assertion made by Xie et al. (Canad. Math. Bull. 56(4) (2013) 881–889). We also extend the improved result for two Heisenberg translations in \(\mathrm{PSp}(2,1)\).

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Acknowledgements

The authors thank the anonymous referee for many helpful comments and suggestions. They also thank Krishnendu Gongopadhyay for suggesting this problem and discussions. The second author (DT) acknowledges the support of ARSI-Foundation.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research (IISER) Pune, Dr Homi Bhabha Road, Ward No. 8, NCL Colony, Pashan, Pune, 411 008, India

    Sagar B Kalane

  2. Bhaskaracharya Pratishthana, Erandwane, Damle Path, Off Law College Road, Pune, 411 004, India

    Devendra Tiwari

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  1. Sagar B Kalane
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  2. Devendra Tiwari
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Correspondence to Sagar B Kalane.

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Communicated by B Sury.

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Kalane, S.B., Tiwari, D. On free group generated by two Heisenberg translations. Proc Math Sci 132, 38 (2022). https://doi.org/10.1007/s12044-022-00691-6

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  • DOI: https://doi.org/10.1007/s12044-022-00691-6

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