John Parker, PhD University of Cambridge

Personal web page

Contact John Parker (email at j.r.parker@durham.ac.uk)

Research Groups

Department of Mathematical Sciences

• Pure Mathematics
• Pure Mathematics: Geometry

Research Interests

• Discrete groups
• Hyperbolic geometry

Publications

Chapter in book

• Parker, John R & Series, Caroline (2004). The mapping class group of the twice punctured torus. In Groups: Topological, Combinatorial and Arithmetic Aspects. Mueller, Thomas W. Cambridge University Press. 311: 405-486.
• Menzel, Christof. & Parker, John R. (2003). Pseudo-Anosov diffeomorphisms of the twice punctured torus. In Recent Advances in Group Theory and Low-Dimensional Topology. Cho, Jung Rae. & Mennicke, Jens. Heldermann Verlag. 27: 141-154.
• Parker, John R. (2003). Tetrahedral decomposition of punctured torus bundles. In Kleinian Groups and Hyperbolic 3-Manifolds. Komori, Yohei., Markovic, Vladimir, & Series, Caroline. Cambridge University Press. 299: 275-291.
• Parker, John R. & Parkkonen, Jouni (1998). Coordinates for quasi-Fuchsian punctured torus space. In The Epstein Birthday Schrift. Geometry and Topology Monographs. 1: 451-478.

Journal Article

• Alvarez Lopez, Jesus Antonio, Barral Lijo, Ramon, Hunton, John, Nozawa, Hiraku & Parker, John R (2021). Chaotic Delone Sets. Discrete and Continuous Dynamical Systems 41(8): 3781-3796.
• Akiyoshi, Hirotaka, Ohshika, Ken'ichi, Parker, John R, Sakuma, Makoto & Yoshida, Han (2021). Classification of non-free Kleinian groups generated by two parabolic transformations. Transactions of the American Mathematical Society 374(3): 1765-1814.
• Deraux, Martin, Parker, John R. & Paupert, Julien (2021). New non-arithmetic complex hyperbolic lattices II. Michigan Mathematical Journal 70(1): 133-205.
• Kneiper, Gerhard, Parker, John R & Peyerimhoff, Norbert (2020). Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces. Differential Geometry and its Applications 69: 101605.
• Parker, John R. (2020). Non-arithmetic monodromy of higher hypergeometric functions. Journal d'Analyse Mathematique 142: 41-70.
• Monaghan, Andrew, Parker, John R. & Pratoussevitch, Anna (2019). Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type [m_1,m_2,0]. Journal of the London Mathematical Society 100(2): 545-567.
• Parker, John R, Peyerimhoff, Norbert & Siburg, Karl Friedrich (2018). Minimizing length of billiard trajectories in hyperbolic polygons. Conformal Geometry and Dynamics 22: 315-332.
• Parker, John R. (2018). On cusp regions associated to screw-parabolic maps. Geometriae Dedicata 192(1): 267-294.
• Cao, Wensheng & Parker, John R. (2018). Shimizu’s Lemma for Quaternionic Hyperbolic Space. Computational Methods and Function Theory 18(1): 159-191.
• Parker, John R. & Will, Pierre (2017). A complex hyperbolic Riley slice. Geometry & Topology 21(6): 3391-3451.
• Parker, John R. & Sun, Lijie (2017). Complex Hyperbolic Triangle Groups with 2-fold Symmetry. Proceedings of the International Geometry Center 10(1): 1-21.
• Cano, Angel, Parker, John R. & Seade, José (2016). Action of R-Fuchsian groups on CP2. The Asian Journal of Mathematics 20(3): 449-474.
• Parker, John R., Wang, Jieyan & Xie, Baohua (2016). Complex hyperbolic (3,3,n)-triangle groups. Pacific Journal of Mathematics 280(2): 433-453.
• Deraux, Martin, Parker, John R. & Paupert, Julien (2016). New non-arithmetic complex hyperbolic lattices. Inventiones Mathematicae 203(3): 681-771.
• Boadi, Richard K & Parker, John R (2015). Mostow's lattices and cone metrics on the sphere. Advances in Geometry 15(1): 27-53.
• Gongopadhyay, Krishnendu, Parker, John R. & Parsad, Shiv (2015). On the classification of unitary matrices. Osaka Journal of Mathematics 52(4): 959-993.
• Gongopadhyay, Krishnendu & Parker, John R (2013). Reversible complex hyperbolic isometries. Linear Algebra and its Applications 438(6): 2728-2739.
• Kamiya, Shigeyasu, Parker, John R. & Thompson, James M. (2012). Non-Discrete Complex Hyperbolic Triangle Groups of Type (n,n,infinity;k). Canadian Mathematical Bulletin 55(2): 329-338.
• Deraux, Martin, Parker, John R & Paupert, Julien (2011). Census of the complex hyperbolic sporadic triangle groups. Experimental Mathematics 20(4): 467-586.
• Falbel,Elisha, Francsics,Gabor, Lax, Peter D & Parker, John R (2011). Generators of a Picard modular group in two complex dimensions. Proceedings of the American Mathematical Society 139: 2439-2447.
• Cao, Wensheng. & Parker, John R. (2011). Jørgensen's inequality and collars in n-dimensional quaternionic hyperbolic space. Quarterly Journal of Mathematics 62(3): 523-543.
• Falbel, Elisha, Francsics, Gabor & Parker, John R (2011). The geometry of the Gauss-Picard modular group. Mathematische Annalen 349(2): 459-508.
• Kamiya, Shigeyasu, Parker, John R. & Thompson, James M. (2010). Notes on complex hyperbolic triangle groups. Conformal Geometry and Dynamics 14: 202-218.
• Parker, John R. & Short, Ian. (2009). Conjugacy classification of quaternionic Moebius transformations. Computational Methods and Function Theory 9(1): 13-25.
• Parker, John R. & Platis, Ioannis D. (2009). Global, geometrical coordinates on Falbel's cross-ratio variety. Canadian Mathematical Bulletin 52(2): 285-294.
• Parker, John R. & Paupert, Julien (2009). Unfaithful complex hyperbolic triangle groups II: Higher order reflections. Pacific Journal of Mathematics 239(2): 357-389.
• Parker,J.R. & Platis,I.D. (2008). Complex hyperbolic Fenchel-Nielsen coordinates. Topology 47(2): 101-135.
• Kamiya,S. & Parker,J.R. (2008). Discrete subgroups of PU(2,1) with screw parabolic elements. Mathematical Proceedings of the Cambridge Philosophical Society 144(2): 443-455.
• Parker, John R. (2008). Unfaithful complex hyperbolic triangle groups I: Involutions. Pacific Journal of Mathematics 238(1): 145-169.
• Markham,Sarah. & Parker,John R. (2007). Jorgensen's inequality for metric spaces with applications to the octonions. Advances in Geometry 7: 19-38.
• Parker, John R. (2006). Cone metrics on the sphere and Livné's lattices. Acta Mathematica 196(1): 1-64.
• Parker, John R & Platis, Ioannis D (2006). Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space. Journal of Differential Geometry 73(2): 319-350.
• Falbel, Elisha & Parker, John R (2006). The geometry of the Eisenstein-Picard modular group. Duke Mathematical Journal 131(2): 249-289.
• Mednykh, Alexander D., Parker, John R. & Vesnin, Andrei Yu (2004). On hyperbolic polyhedra arising as convex cores of quasi-Fuchsian punctured torus groups. Boletin de la Sociedad Matematica Mexicana 10(3): 357-381.
• Cao, Wensheng, Parker, John R. & Wang, Xiantao (2004). On the classification of quaternionic Moebius transformations. Mathematical Proceedings of the Cambridge Philosophical Society 137(2): 349-361.
• Markham, Sarah. & Parker, John R. (2003). Collars in complex and quaternionic hyperbolic manifolds. Geometriae Dedicata 97: 199-213.
• Gusevskii, Nikolay & Parker, John R. (2003). Complex hyperbolic quasi-Fuchsian groups and Toledo's invariant. Geometriae Dedicata 97: 151-185.
• Kim, Inkang & Parker, John R (2003). Geometry of quaternionic hyperbolic manifolds. Mathematical Proceedings of the Cambridge Philosophical Society 135(2): 291-320.
• Jiang, Yueping, Kamiya, Shigeyasu & Parker, John R (2003). Jorgensen's inequality for complex hyperbolic space. Geometriae Dedicata 97: 55-80.
• Falbel, Elisha & Parker, John R. (2003). The moduli space of the modular group in complex hyperbolic geometry. Inventiones Mathematicae 152(1): 57-88.
• Jiang, Yueping & Parker, John R. (2003). Uniform discreteness and Heisenberg scew motions. Mathematische Zeitschrift 243: 653-669.
• Kamiya, Shigeyasu. & Parker, John R. (2002). On discrete subgroups of PU(1,2;C) with Heisenberg translations II. Revue Roumaine Mathematiques Pures et Appliquees 47(5-6): 689-695.
• Parker, John R. & Stratmann, Bernd O. (2001). Kleinian groups with singly cusped parabolic fixed points. Kodai Mathematical Journal 24(2): 169-206.
• Gusevskii, Nikolay & Parker, John R. (2000). Representations of free Fuchsian groups in complex hyperbolic space. Topology 39(1): 33-60.
• Keen, Linda, Parker, John R. & Series, Caroline (1999). Combinatorics of simple closed curves on the twice punctured torus. Israel Journal of Mathematics 112: 29-60.
• Parker, John R. (1998). On the volumes of cusped, complex hyperbolic manifolds and orbifolds. Duke Mathematical Journal 94(3): 433-464.
• Parker, John R. (1997). Uniform discreteness and Heisenberg translations. Mathematische Zeitschrift 225: 484-505.
• Parker, John R. & Series, Caroline. (1996). Bending Formulae for Convex Hull Boundaries. Journal d'Analyse Mathematique 67: 165-198.
• Parker, John R. (1995). Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space. Geometriae Dedicata 57(3): 223-234.
• Parker, John R. (1995). Kleinian circle packings. Topology 34(3): 489-496.
• McShane, Greg., Parker, John R. & Redfern, Ian (1994). Drawing Limit Sets of Kleinian Groups using Finite State Automata. Experimental Mathematics 3: 153-170.
• Parker, John R. (1994). On Ford Isometric Spheres in Complex Hyperbolic Space. Mathematical Proceedings of the Cambridge Philosophical Society 115: 501-512.

Conference Paper

• Parker, John R. & Will, Pierre (2015), Complex hyperbolic free groups with many parabolic elements, Contemporary Mathematics 639: Groups, Geometry and Dynamics. Almora, Uttarakhand, India, American Mathematical Society, 327-348.
• Parker, John R. (2012), Traces in complex hyperbolic geometry, in Goldman, William M., Series, Caroline & Tan, Ser Peow eds, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 23: Geometry, Topology and Dynamics of Character Varieties. Singapore, World Scientific, Singapore, 191-245.
• Parker, John R. & Platis, Ioannis D. (2010), Complex hyperbolic quasi-Fuchsian groups, in Gardiner, Frederick P. González-Diez, Gabino & Kourouniotis, Christos eds, 368: Geometry of Riemann Surfaces. Anogia, Crete, Greece, Cambridge University Press, Cambridge, 309-355.
• Parker, John R. (2009), Complex hyperbolic lattices, in Dekimpe, Karel, Igodt, Paul & Valette, Alain eds, 501: Discrete groups and geometric structures. Kortrijk, Belgium, American Mathematical Society, Providence, R.I., 1-42.

Conference Proceeding

• Armitage, J.Vernon. & Parker, John R. (2008). Jorgensen's inequality for non-Archimedean metric spaces. Geometry and Dynamics of Groups and Spaces In Memory of Alexander Reznikov, Bonn, Birkhauser.

Supervises

• Mr Wanchalerm Promduang

Indicators of Esteem

• 'Grants':

  'Research grants:
  

  JSPS Invitation Fellowship 2016.

  UK-Mexico Visiting Chair 2017

• 'Invitation to research centres': Tokyo Institute of Technology, April 2016 to March 2017 (supported by JSPS).
• 'National and International Collaboration':

  'International collaborators:

  RK Boadi, KNUST, Ghana

  W Cao, Hunan University of Science and Technology, China

  A Cano, UNAM, Mexico

  M Deraux, Fourier Institute, Grenoble, France

  E Falbel, University of Paris VI, France

  N Gusevskii, Federal University of Minas Gerais, Brazil

  Y Jiang, Hunan University, China

  S Kamiya, Okayama University of Science, Japan

  I Kim, Seoul National University, Korea

  A D Mednykh, Sobelev Institute, Russia

  C Menzel, University of Bielefeld, Germany

  J Paupert, Arizona State University, USA

  I D Platis, University of Crete

  J Seade, UNAM, Mexico

  A Y Vesnin, Sobelev Institute, Russia

  J Wang, Hunan University, China

  X Wang, Hunan Normal University, China

  P Will, Fourier Institute, Grenoble, France

  B Xie, Hunan University, China

• Editorial work: Joint Editor-in-Chief, Geometriae Dedicata