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Conformal Geometry and Dynamics

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Complex hyperbolic triangle groups of type $ [m,m,0;3,3,2]$


Authors: Sam Povall and Anna Pratoussevitch
Journal: Conform. Geom. Dyn. 24 (2020), 51-67
MSC (2010): Primary 51M10; Secondary 32M15, 22E40, 53C55
DOI: https://doi.org/10.1090/ecgd/348
Published electronically: February 13, 2020
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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.


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Additional Information

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: sam.povall@unimelb.edu.au

Anna Pratoussevitch
Affiliation: Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, United Kingdom
Email: annap@liverpool.ac.uk

DOI: https://doi.org/10.1090/ecgd/348
Keywords: Complex hyperbolic geometry, triangle groups
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
Article copyright: © Copyright 2020 American Mathematical Society