Abstract
Let A and B be two unipotent elements of
Funding source: National Natural Science Foundation of China
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
[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
[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
[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
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Articles in the same Issue
- Frontmatter
- Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
- Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
- Estimates of Picard modular cusp forms
- Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
- Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
- Free groups generated by two unipotent maps
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- Simultaneous nonvanishing of central L-values with large level
- Laplace convolutions of weighted averages of arithmetical functions
- Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
- On the geometric trace of a generalized Selberg trace formula
- Elementary properties of free lattices
- Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
- Small generators of abelian number fields
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Articles in the same Issue
- Frontmatter
- Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
- Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
- Estimates of Picard modular cusp forms
- Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
- Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
- Free groups generated by two unipotent maps
- Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
- Colored multizeta values in positive characteristic
- Simultaneous nonvanishing of central L-values with large level
- Laplace convolutions of weighted averages of arithmetical functions
- Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
- On the geometric trace of a generalized Selberg trace formula
- Elementary properties of free lattices
- Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
- Small generators of abelian number fields
- Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
- Weighted bilinear multiplier theorems in Dunkl setting via singular integrals