Complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$
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- by Sam Povall and Anna Pratoussevitch PDF
- Conform. Geom. Dyn. 24 (2020), 51-67 Request permission
Abstract:In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e., groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ in complex geodesics with pairwise distances $m,m,0$. For fixed $m$, the parameter space of such groups is of real dimension one. We determine intervals in this parameter space that correspond to discrete and to non-discrete triangle groups.
- Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek, and Hans Zassenhaus, Crystallographic groups of four-dimensional space, Wiley Monographs in Crystallography, Wiley-Interscience [John Wiley & Sons], New York-Chichester-Brisbane, 1978. MR 0484179
- Karel Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. MR 1482520, DOI 10.1007/BFb0094472
- William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450
- Andrew Monaghan, Complex hyperbolic triangle groups, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–The University of Liverpool (United Kingdom). MR 3389497
- Andrew Monaghan, John R. Parker, and Anna Pratoussevitch, Discreteness of ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, J. Lond. Math. Soc. (2) 100 (2019), no. 2, 545–567. MR 4017154, DOI 10.1112/jlms.12227
- John R. Parker, Shimizu’s lemma for complex hyperbolic space, Internat. J. Math. 3 (1992), no. 2, 291–308. MR 1146815, DOI 10.1142/S0129167X92000096
- John R. Parker, On Ford isometric spheres in complex hyperbolic space, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 501–512. MR 1269935, DOI 10.1017/S0305004100072261
- John R. Parker, Uniform discreteness and Heisenberg translations, Math. Z. 225 (1997), no. 3, 485–505. MR 1465903, DOI 10.1007/PL00004315
- J. R. Parker, Notes on complex hyperbolic geometry, lecture notes, 2010.
- S. Povall, Ultra-parallel complex hyperbolic triangle groups, Ph.D. thesis, University of Liverpool, 2019.
- Anna Pratoussevitch, Traces in complex hyperbolic triangle groups, Geom. Dedicata 111 (2005), 159–185. MR 2155180, DOI 10.1007/s10711-004-1493-0
- Richard Evan Schwartz, Complex hyperbolic triangle groups, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 339–349. MR 1957045
- Justin Olav Wyss-Gallifent, Complex hyperbolic triangle groups, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Ph.D.)–University of Maryland, College Park. MR 2700554
- Sam Povall
- Affiliation: Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom.
- Address at time of publication: Department of Mathematics & Statistics, University of Melbourne, Parkville, Victoria, 3052, Australia
- Email: email@example.com
- Anna Pratoussevitch
- Affiliation: Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom
- MR Author ID: 704274
- Email: firstname.lastname@example.org
- Received by editor(s): March 3, 2019
- Received by editor(s) in revised form: September 16, 2019
- Published electronically: February 13, 2020
- Additional Notes: The first author acknowledges financial support from an EPSRC DTA scholarship at the University of Liverpool and also partial support by the International Centre for Theoretical Sciences (ICTS) during the participation in the programmes Geometry, Groups and Dynamics (ICTS/ggd2017/11) and Surface Group Representations and Geometric Structures (ICTS/SGGS2017/11).
The second author also acknowledges support from the ICTS
- © Copyright 2020 American Mathematical Society
- Journal: Conform. Geom. Dyn. 24 (2020), 51-67
- MSC (2010): Primary 51M10; Secondary 32M15, 22E40, 53C55
- DOI: https://doi.org/10.1090/ecgd/348
- MathSciNet review: 4063328