Staff profile

• discrete groups
• hyperbolic geometry

• 2000: Vice-Chair HoDoMS (Heads of Departments of Mathematical Sciences):
• 2000: '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

• 2000: 'Grants': 'Research grants:
  

  JSPS Invitation Fellowship 2016.

  UK-Mexico Visiting Chair 2017

• 2000: Editorial work: Joint Editor-in-Chief, Geometriae Dedicata
• 2000: 'Invitation to research centres': Tokyo Institute of Technology, April 2016 to March 2017 (supported by JSPS).

Chapter in book

• The mapping class group of the twice punctured torus
  Parker, J. R., & Series, C. (2004). The mapping class group of the twice punctured torus. In T. W. Mueller (Ed.), Groups: Topological, Combinatorial and Arithmetic Aspects (405-486). (LMS Lecture Notes). Cambridge University Press
• Tetrahedral decomposition of punctured torus bundles
  Parker, J. R. (2003). Tetrahedral decomposition of punctured torus bundles. In Y. Komori, V. Markovic, & C. Series (Eds.), Kleinian Groups and Hyperbolic 3-Manifolds (275-291). (LMS Lecture Notes). Cambridge University Press
• Pseudo-Anosov diffeomorphisms of the twice punctured torus
  Menzel, C., & Parker, J. R. (2003). Pseudo-Anosov diffeomorphisms of the twice punctured torus. In J. R. Cho, & J. Mennicke (Eds.), Recent Advances in Group Theory and Low-Dimensional Topology (141-154). (Research and Exposition in Mathematics). Heldermann Verlag
• Coordinates for quasi-Fuchsian punctured torus space,
  Parker, J. R., & Parkkonen, J. (1998). Coordinates for quasi-Fuchsian punctured torus space,. In The Epstein Birthday Schrift (451-478). Geometry and Topology Monographs

Conference Paper

• Complex hyperbolic free groups with many parabolic elements
  Parker, J. R., & Will, P. (2012, December). Complex hyperbolic free groups with many parabolic elements. Presented at Groups, Geometry and Dynamics, Almora, Uttarakhand, India
• Traces in complex hyperbolic geometry
  Parker, J. R. (2012, August). Traces in complex hyperbolic geometry. Presented at Geometry, Topology and Dynamics of Character Varieties, National University of Singapore
• Complex hyperbolic quasi-Fuchsian groups
  Parker, J. R., & Platis, I. D. (2010, February). Complex hyperbolic quasi-Fuchsian groups. Presented at Geometry of Riemann Surfaces, Anogia, Crete, Greece
• Complex hyperbolic lattices
  Parker, J. R. (2009, December). Complex hyperbolic lattices. Presented at Discrete groups and geometric structures, Kortrijk, Belgium
• Jorgensen's inequality for non-Archimedean metric spaces
  Armitage, J., & Parker, J. R. (2006, September). Jorgensen's inequality for non-Archimedean metric spaces. Presented at Geometry and dynamics of groups and spaces : in memory of Alexander Reznikov., Bonn, Germany

Journal Article

• Fenchel-Nielsen coordinates for SL(3,C) representations
  Davila Figueroa, R., & Parker, J. R. (2025). Fenchel-Nielsen coordinates for SL(3,C) representations. Geometriae Dedicata, 219, Article 63
• Free groups generated by two parabolic maps
  Kalane, S. B., & Parker, J. R. (2023). Free groups generated by two parabolic maps. Mathematische Zeitschrift, 303, Article 9. https://doi.org/10.1007/s00209-022-03160-y
• Chaotic Delone Sets
  Alvarez Lopez, J. A., Barral Lijo, R., Hunton, J., Nozawa, H., & Parker, J. R. (2021). Chaotic Delone Sets. Discrete and Continuous Dynamical Systems - Series A, 41(8), 3781-3796. https://doi.org/10.3934/dcds.2021016
• New non-arithmetic complex hyperbolic lattices II
  Deraux, M., Parker, J. R., & Paupert, J. (2021). New non-arithmetic complex hyperbolic lattices II. Michigan Mathematical Journal, 70(1), 133-205. https://doi.org/10.1307/mmj/1592532044
• Classification of non-free Kleinian groups generated by two parabolic transformations
  Akiyoshi, H., Ohshika, K., Parker, J. R., Sakuma, M., & Yoshida, H. (2021). Classification of non-free Kleinian groups generated by two parabolic transformations. Transactions of the American Mathematical Society, 374(3), 1765-1814. https://doi.org/10.1090/tran/8246
• Non-arithmetic monodromy of higher hypergeometric functions
  Parker, J. R. (2020). Non-arithmetic monodromy of higher hypergeometric functions. Journal d'Analyse Mathématique, 142, 41-70. https://doi.org/10.1007/s11854-020-0132-5
• Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces
  Knieper, G., Parker, J. R., & Peyerimhoff, N. (2020). Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces. Differential Geometry and its Applications, 69, Article 101605. https://doi.org/10.1016/j.difgeo.2020.101605
• Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type [m_1,m_2,0]
  Monaghan, A., Parker, J. R., & Pratoussevitch, A. (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. https://doi.org/10.1112/jlms.12227
• Minimizing length of billiard trajectories in hyperbolic polygons
  Parker, J. R., Peyerimhoff, N., & Siburg, K. F. (2018). Minimizing length of billiard trajectories in hyperbolic polygons. Conformal Geometry and Dynamics, 22, 315-332. https://doi.org/10.1090/ecgd/328
• Shimizu’s Lemma for Quaternionic Hyperbolic Space
  Cao, W., & Parker, J. R. (2018). Shimizu’s Lemma for Quaternionic Hyperbolic Space. Computational Methods and Function Theory - Springer, 18(1), 159-191. https://doi.org/10.1007/s40315-017-0212-4
• On cusp regions associated to screw-parabolic maps
  Parker, J. R. (2018). On cusp regions associated to screw-parabolic maps. Geometriae Dedicata, 192(1), 267-294. https://doi.org/10.1007/s10711-017-0241-1
• A complex hyperbolic Riley slice
  Parker, J. R., & Will, P. (2017). A complex hyperbolic Riley slice. Geometry & Topology, 21(6), 3391-3451. https://doi.org/10.2140/gt.2017.21.3391
• Complex Hyperbolic Triangle Groups with 2-fold Symmetry
  Parker, J. R., & Sun, L. (2017). Complex Hyperbolic Triangle Groups with 2-fold Symmetry. Proceedings of the International Geometry Center, 10(1), 1-21. https://doi.org/10.15673/tmgc.v1i10.547
• Action of R-Fuchsian groups on CP2
  Cano, A., Parker, J. R., & Seade, J. (2016). Action of R-Fuchsian groups on CP2. Asian Journal of Mathematics, 20(3), 449-474. https://doi.org/10.4310/ajm.2016.v20.n3.a3
• New non-arithmetic complex hyperbolic lattices
  Deraux, M., Parker, J. R., & Paupert, J. (2016). New non-arithmetic complex hyperbolic lattices. Inventiones Mathematicae, 203(3), 681-771. https://doi.org/10.1007/s00222-015-0600-1
• Complex hyperbolic (3,3,n)-triangle groups
  Parker, J. R., Wang, J., & Xie, B. (2016). Complex hyperbolic (3,3,n)-triangle groups. Pacific Journal of Mathematics, 280(2), 433-453. https://doi.org/10.2140/pjm.2016.280.433
• On the classification of unitary matrices
  Gongopadhyay, K., Parker, J. R., & Parsad, S. (2015). On the classification of unitary matrices. Osaka Journal of Mathematics, 52(4), 959-993
• Mostow's lattices and cone metrics on the sphere
  Boadi, R. K., & Parker, J. R. (2015). Mostow's lattices and cone metrics on the sphere. Advances in Geometry, 15(1), 27-53. https://doi.org/10.1515/advgeom-2014-0022
• Reversible complex hyperbolic isometries
  Gongopadhyay, K., & Parker, J. R. (2013). Reversible complex hyperbolic isometries. Linear Algebra and its Applications, 438(6), 2728-2739. https://doi.org/10.1016/j.laa.2012.11.029
• Non-Discrete Complex Hyperbolic Triangle Groups of Type (n, n, ∞; k)
  Kamiya, S., Parker, J. R., & Thompson, J. M. (2012). Non-Discrete Complex Hyperbolic Triangle Groups of Type (n, n, ∞; k). Canadian Mathematical Bulletin, 55(2), 329-338. https://doi.org/10.4153/cmb-2011-094-8
• Census of the complex hyperbolic sporadic triangle groups
  Deraux, M., Parker, J. R., & Paupert, J. (2011). Census of the complex hyperbolic sporadic triangle groups. Experimental Mathematics, 20(4), 467-586. https://doi.org/10.1080/10586458.2011.565262
• The geometry of the Gauss-Picard modular group
  Falbel, E., Francsics, G., & Parker, J. R. (2011). The geometry of the Gauss-Picard modular group. Mathematische Annalen, 349(2), 459-508. https://doi.org/10.1007/s00208-010-0515-5
• Generators of a Picard modular group in two complex dimensions
  Falbel, E., Francsics, G., Lax, P. D., & Parker, J. R. (2011). Generators of a Picard modular group in two complex dimensions. Proceedings of the American Mathematical Society, 139, 2439-2447. https://doi.org/10.1090/s0002-9939-2010-10653-6
• Jørgensen's inequality and collars in n-dimensional quaternionic hyperbolic space.
  Cao, W., & Parker, J. R. (2011). Jørgensen's inequality and collars in n-dimensional quaternionic hyperbolic space. The Quarterly Journal of Mathematics, 62(3), 523-543. https://doi.org/10.1093/qmath/haq003
• Notes on complex hyperbolic triangle groups
  Kamiya, S., Parker, J. R., & Thompson, J. M. (2010). Notes on complex hyperbolic triangle groups. Conformal Geometry and Dynamics, 14, 202-218. https://doi.org/10.1090/s1088-4173-2010-00215-8
• Unfaithful complex hyperbolic triangle groups II: Higher order reflections
  Parker, J. R., & Paupert, J. (2009). Unfaithful complex hyperbolic triangle groups II: Higher order reflections. Pacific Journal of Mathematics, 239(2), 357-389. https://doi.org/10.2140/pjm.2009.239.357
• Global, geometrical coordinates on Falbel's cross-ratio variety
  Parker, J. R., & Platis, I. D. (2009). Global, geometrical coordinates on Falbel's cross-ratio variety. Canadian Mathematical Bulletin, 52(2), 285-294. https://doi.org/10.4153/cmb-2009-031-3
• Conjugacy classification of quaternionic Möbius transformations
  Parker, J. R., & Short, I. (2009). Conjugacy classification of quaternionic Möbius transformations. Computational Methods and Function Theory - Springer, 9(1), 13-25. https://doi.org/10.1007/bf03321711
• Unfaithful complex hyperbolic triangle groups I: Involutions
  Parker, J. R. (2008). Unfaithful complex hyperbolic triangle groups I: Involutions. Pacific Journal of Mathematics, 238(1), 145-169. https://doi.org/10.2140/pjm.2008.238.145
• Complex hyperbolic Fenchel-Nielsen coordinates
  Parker, J., & Platis, I. (2008). Complex hyperbolic Fenchel-Nielsen coordinates. Topology (Oxford), 47(2), 101-135. https://doi.org/10.1016/j.top.2007.08.001
• Discrete subgroups of PU(2,1) with screw parabolic elements
  Kamiya, S., & Parker, J. (2008). Discrete subgroups of PU(2,1) with screw parabolic elements. Mathematical Proceedings of the Cambridge Philosophical Society, 144(2), 443-455. https://doi.org/10.1017/s0305004107000941
• Jorgensen's inequality for metric spaces with applications to the octonions.
  Markham, S., & Parker, J. R. (2007). Jorgensen's inequality for metric spaces with applications to the octonions. Advances in Geometry, 7, 19-38. https://doi.org/10.1515/advgeom.2007.002
• Cone metrics on the sphere and Livne's lattices
  Parker, J. R. (2006). Cone metrics on the sphere and Livne's lattices. Acta Mathematica, 196(1), 1-64. https://doi.org/10.1007/s11511-006-0001-9
• Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space
  Parker, J. R., & Platis, I. D. (2006). Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space. Journal of Differential Geometry, 73(2), 319-350
• The geometry of the Eisenstein-Picard modular group
  Falbel, E., & Parker, J. R. (2006). The geometry of the Eisenstein-Picard modular group. Duke Mathematical Journal, 131(2), 249-289. https://doi.org/10.1215/s0012-7094-06-13123-x
• On the classification of quaternionic Moebius transformations
  Cao, W., Parker, J. R., & Wang, X. (2004). On the classification of quaternionic Moebius transformations. Mathematical Proceedings of the Cambridge Philosophical Society, 137(2), 349-361. https://doi.org/10.1017/s0305004104007868
• On hyperbolic polyhedra arising as convex cores of quasi-Fuchsian punctured torus groups.
  Mednykh, A. D., Parker, J. R., & Vesnin, A. Y. (2004). On hyperbolic polyhedra arising as convex cores of quasi-Fuchsian punctured torus groups. Boletín de la Sociedad Matemática Mexicana, 10(3), 357-381
• The moduli space of the modular group in complex hyperbolic geometry
  Falbel, E., & Parker, J. R. (2003). The moduli space of the modular group in complex hyperbolic geometry. Inventiones Mathematicae, 152(1), 57-88. https://doi.org/10.1007/s00222-002-0267-2
• Complex hyperbolic quasi-Fuchsian groups and Toledo's invariant
  Gusevskii, N., & Parker, J. R. (2003). Complex hyperbolic quasi-Fuchsian groups and Toledo's invariant. Geometriae Dedicata, 97, 151-185
• Geometry of quaternionic hyperbolic manifolds
  Kim, I., & Parker, J. R. (2003). Geometry of quaternionic hyperbolic manifolds. Mathematical Proceedings of the Cambridge Philosophical Society, 135(2), 291-320. https://doi.org/10.1017/s030500410300687x
• Collars in complex and quaternionic hyperbolic manifolds
  Markham, S., & Parker, J. R. (2003). Collars in complex and quaternionic hyperbolic manifolds. Geometriae Dedicata, 97, 199-213
• Uniform discreteness and Heisenberg scew motions
  Jiang, Y., & Parker, J. R. (2003). Uniform discreteness and Heisenberg scew motions. Mathematische Zeitschrift, 243, 653-669
• Jorgensen's inequality for complex hyperbolic space
  Jiang, Y., Kamiya, S., & Parker, J. R. (2003). Jorgensen's inequality for complex hyperbolic space. Geometriae Dedicata, 97, 55-80
• On discrete subgroups of PU(1,2;C) with Heisenberg translations II
  Kamiya, S., & Parker, J. R. (2002). On discrete subgroups of PU(1,2;C) with Heisenberg translations II
• Kleinian groups with singly cusped parabolic fixed points
  Parker, J. R., & Stratmann, B. O. (2001). Kleinian groups with singly cusped parabolic fixed points
• Representations of free Fuchsian groups in complex hyperbolic space
  Gusevskii, N., & Parker, J. R. (2000). Representations of free Fuchsian groups in complex hyperbolic space
• Combinatorics of simple closed curves on the twice punctured torus
  Keen, L., Parker, J. R., & Series, C. (1999). Combinatorics of simple closed curves on the twice punctured torus. Israel Journal of Mathematics, 112, 29-60
• On the volumes of cusped, complex hyperbolic manifolds and orbifolds
  Parker, J. R. (1998). On the volumes of cusped, complex hyperbolic manifolds and orbifolds. Duke Mathematical Journal, 94(3), 433-464. https://doi.org/10.1215/s0012-7094-98-09418-2
• Uniform discreteness and Heisenberg translations
  Parker, J. R. (1997). Uniform discreteness and Heisenberg translations. Mathematische Zeitschrift, 225, 484-505
• Bending Formulae for Convex Hull Boundaries
  Parker, J. R., & Series, C. (1996). Bending Formulae for Convex Hull Boundaries. Journal d'Analyse Mathématique, 67, 165-198
• Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space
  Parker, J. R. (1995). Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space. Geometriae Dedicata, 57(3), 223-234
• Kleinian circle packings
  Parker, J. R. (1995). Kleinian circle packings
• Drawing Limit Sets of Kleinian Groups using Finite State Automata
  McShane, G., Parker, J. R., & Redfern, I. (1994). Drawing Limit Sets of Kleinian Groups using Finite State Automata. Experimental Mathematics, 3, 153-170
• On Ford Isometric Spheres in Complex Hyperbolic Space
  Parker, J. R. (1994). On Ford Isometric Spheres in Complex Hyperbolic Space. Mathematical Proceedings of the Cambridge Philosophical Society, 115, 501-512