Skip to content
Licensed Unlicensed Requires Authentication Published online by De Gruyter March 26, 2024

Free groups generated by two unipotent maps

  • Chao Jiang and Baohua Xie ORCID logo EMAIL logo
From the journal Forum Mathematicum

Abstract

Let A and B be two unipotent elements of SU ( 2 , 1 ) with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup A , B is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that AB is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup A , B is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].

MSC 2020: 22E40; 51M10; 20H10

Communicated by Manfred Droste


Award Identifier / Grant number: 11871202

Award Identifier / Grant number: 12271148

Funding statement: This work was supported by the National Natural Science Foundation of China (Grants No. 11871202, No. 12271148).

Acknowledgements

We would like to thank the anonymous referee, whose insightful suggestions helped improving earlier versions of the manuscript. We thank Wei Liao and Mengqi Yu for several useful discussions.

References

[1] A. F. Beardon, The Geometry of Discrete Groups, Grad. Texts in Math. 91, Springer, New York, 1983. 10.1007/978-1-4612-1146-4Search in Google Scholar

[2] J. L. Brenner, R. A. MacLeod and D. D. Olesky, Non-free groups generated by two 2 × 2 matrices, Canad. J. Math. 27 (1975), 237–245. 10.4153/CJM-1975-029-5Search in Google Scholar

[3] E. Falbel, G. Francsics and J. R. Parker, The geometry of the Gauss-Picard modular group, Math. Ann. 349 (2011), no. 2, 459–508. 10.1007/s00208-010-0515-5Search in Google Scholar

[4] E. Falbel and J. R. Parker, The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131 (2006), no. 2, 249–289. 10.1215/S0012-7094-06-13123-XSearch in Google Scholar

[5] J. Gilman, The structure of two-parabolic space: Parabolic dust and iteration, Geom. Dedicata 131 (2008), 27–48. 10.1007/s10711-007-9215-zSearch in Google Scholar

[6] W. M. Goldman, Complex Hyperbolic Geometry, Oxford Math. Monogr., Clarendon Press, Oxford, 1999. 10.1093/oso/9780198537939.001.0001Search in Google Scholar

[7] J. A. Ignatov, Free and nonfree subgroups of PSL 2 ( 𝐂 ) that are generated by two parabolic elements, Mat. Sb. (N. S.) 106(148) (1978), no. 3, 372–379. Search in Google Scholar

[8] S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 (2023), no. 1, Paper No. 9. 10.1007/s00209-022-03160-ySearch in Google Scholar

[9] S. Kamiya, On discrete subgroups of PU ( 1 , 2 ; 𝐂 ) with Heisenberg translations, J. Lond. Math. Soc. (2) 62 (2000), 827–842. 10.1112/S0024610700001435Search in Google Scholar

[10] W. Liao and B. H. Xie, Free groups generated by two screw parabolic maps, preprint. Search in Google Scholar

[11] R. C. Lyndon and J. L. Ullman, Groups generated by two parabolic linear fractional transformations, Canad. J. Math. 21 (1969), 1388–1403. 10.4153/CJM-1969-153-1Search in Google Scholar

[12] J. R. Parker, Uniform discreteness and Heisenberg translations, Math. Z. 225 (1997), no. 3, 485–505. 10.1007/PL00004315Search in Google Scholar

[13] J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 (2017), no. 6, 3391–3451. 10.2140/gt.2017.21.3391Search in Google Scholar

[14] M. B. Phillips, Dirichlet polyhedra for cyclic groups in complex hyperbolic space, Proc. Amer. Math. Soc. 115 (1992), no. 1, 221–228. 10.1090/S0002-9939-1992-1107276-1Search in Google Scholar

[15] R. Riley, A personal account of the discovery of hyperbolic structures on some knot complements, Expo. Math. 31 (2013), no. 2, 104–115. 10.1016/j.exmath.2013.01.003Search in Google Scholar

[16] B. Xie, J. Wang and Y. Jiang, Free groups generated by two Heisenberg translations, Canad. Math. Bull. 56 (2013), no. 4, 881–889. 10.4153/CMB-2012-042-0Search in Google Scholar

Received: 2023-12-05
Published Online: 2024-03-26

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 2.5.2024 from https://www.degruyter.com/document/doi/10.1515/forum-2023-0442/html
Scroll to top button