Korean J. Math. Vol. 30 No. 2 (2022) pp.315-333
DOI: https://doi.org/10.11568/kjm.2022.30.2.315

C-Fuchsian subgroups of some non-arithmetic lattices

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Published: Jun 30, 2022
Keywords:
C-Fuchsian subgroup, Complex hyperbolic triangle group, Poincare's polygon theorem
Mathematics Subject Classification:
32M15, 51M10, 22E40, 20F05

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Lijie Sun
Department of applied science, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan.

Abstract

We give a general procedure to analyze the structure for certain $\mathbb{C}$-Fuchsian subgroups of some non-arithmetic lattices. We also show their presentations and describe their fundamental domains which lie in a complex geodesic, a set homeomorphic to the unit disk.



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References

[1] A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, 1983. Google Scholar

[2] M. Deraux, Deforming the R-Fuchsian (4, 4, 4)-triangle group into a lattice. Topology, 45 (2006), 989–1020. Google Scholar

[3] M. Deraux, Non-arithmetic lattices and the klein quartic, J. Reine Angew. Math., 754 (2019), 253–279. Google Scholar

[4] M. Deraux, J. R. Parker, and J. Paupert, New non-arithmetic complex hyperbolic lattices, Invent. Math., 203 (2015), 681–771. Google Scholar

[5] M. Deraux, J. R. Parker, and J. Paupert, New non-arithmetic complex hyperbolic lattices II, Michigan Math. J., 70 (2021), 135–205. Google Scholar

[6] W. M. Goldman, Complex hyperbolic geometry, Oxford University Press, 1999. Google Scholar

[7] N. Gusevskii and J. R. Parker, Representations of free fuchsian groups in complex hyperbolic space, Topology, 39 (2000), 33–60. Google Scholar

[8] G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math., 86 (1980), 171–276. Google Scholar

[9] J. R. Parker, Unfaithful complex hyperbolic triangle groups. I. Involutions, Pacific J. Math., 238(2008), 145–169. Google Scholar

[10] J. R. Parker and J. Paupert, Unfaithful complex hyperbolic triangle groups. II. Higher order reflections, Pacific J. Math., 239 (2009), 357–389. Google Scholar

[11] J. R. Parker and I. D. Platis, Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space, J. Differential Geom., 73 (2006), 319–350. Google Scholar

[12] J. Parkkonen and F. Paulin, A classification of C-Fuchsian subgroups of Picard modular groups, Math. Scand., 121 (2017), 57–74. Google Scholar

[13] R. E. Schwartz, Real hyperbolic on the outside, complex hyperbolic on the inside, Invent. Math., 151 (2003), 221–295. Google Scholar

[14] M. Stover, Arithmeticity of complex hyperbolic triangle groups, Pacific Journal of Mathematics, 257 (2012), 243–256. Google Scholar

[15] J. M. Thompson, Complex hyperbolic triangle groups, Doctoral thesis, 2010. Google Scholar

[16] J. Wells, Non-arithmetic hybrid lattices in PU(2, 1), Geom. Dedicata, 208 (2020), 1–11. Google Scholar