John R. Parker's Home Page

University of Durham --- Department of Mathematical Sciences

Professor John R. Parker

Department of Mathematical Sciences
Durham University
Science Laboratories
South Road
Durham, DH1 3LE
United Kingdom

Phone:: 0191 334 3057
Department fax:: 0191 334 3051
e-mail:: j.r.parker@durham.ac.uk
Google scholar profile
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Research

Events

Hyperbolic geometry

I work in this general area. Our department logo at the top right hand corner of this page depicts the Poincaré disc model of the hyperbolic plane together with the fundamental domain for an ideal triangle group and several of its images. It may be found on the floor of the entrance to the department. The group is generated by reflection across the sides of the triangle. This group is rigid in the sense that any two such groups are conjugate via a Möbius transformation. On the other hand if we ask an analogous question for complex hyperbolic space there is a one parameter family of inequivalent groups, see the paper (with Bill Goldman) in J. reine angewandte Math in my publications list.

At the moment I mostly work on :

Complex hyperbolic geometry Many of the questions that can be asked for discrete groups of (real) hyperbolic isometries can be asked for complex hyperbolic isometries. However the answers can often be surprising and usually involve rather different methods. This means that complex hyperbolic geometry is a rather exciting field to work in.

Geometriae Dedicata

Between January 2013 and December 2022 Jean-Marc Schlenker and I were joint Editors-in-Chief of Geometriae Dedicata. From January 2023 Richard Wentworth has taken over from us as Editor-in-Chief. We would welcome the the submission of good papers in the area of geometry and its relationship to topology, group theory and the theory of dynamical systems. Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and related areas.

Preprints

with Rodrigo Dávila Figueroa
Fenchel-Nielsen coordinates for SL(3,C) representations.

Publications

Note: The papers listed here are not all linked to retrievable files. And those that are are not necessarily the final versions and so they should be treated as "for information only".

Caroline Series and Hyperbolic Geometry
with Ser Peow Tan
Notices of the American Mathematical Society 70, No 3(2023) 380-389.

with Sagar Kalane
Free groups generated by two parabolic maps
Mathematische Zeitschrift 303 (2023).

with Jesús Antonio Álvarez López, Ramón Barral Lijó, John Hunton and Hiraku Nozawa :
Chaotic Delone sets
Discrete and Continuous Dynamical Systems 41 (2021) 3781-3796.

with Martin Deraux and Julien Paupert:
New non-arithmetic complex hyperbolic lattices II
Martin Deraux's software used in this project spocheck.
Michigan Mathematical Journal 70 (2021) 133-205.

with Hirotaka Akiyoshi, Ken'ichi Ohshika, Makoto Sakuma and Han Yoshida:
Classification of non-free Kleinian groups generated by two parabolic transformations
Transactions of the American Mathematical Society 374 (2021) 1765-1814.

Non-arithmetic monodromy of higher hypergeometric functions
Journal d'Analyse Mathématique 142 (2020) 41-70.

with Gerhard Knieper and and Norbert Peyerimhoff:
Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces
Differential Geometry and its Applications 69 (2020) 101605.

with Andrew Monaghan and and Anna Pratoussevitch:
Discreteness of ultra-parallel complex hyperbolic triangle groups of type [m₁,m₂,0]
Journal of the London Mathematical Society 100 (2019) 545-567.

with Norbert Peyerimhoff and Karl Friedrich Siburg:
Minimizing length of billiard trajectories in hyperbolic polygons
Conformal Geometry and Dynamics 22 (2018) 315-332.

with Wensheng Cao:
Shimizu's lemma for quaternionic hyperbolic space
Computational Methods and Function Theory 18 (2018) 159-191.

On cusp regions associated to screw-parabolic maps
Geometriae Dedicata 192 (2018) 267-294.

with Lijie Sun:
Complex hyperbolic triangle groups with 2-fold symmetry
Proceedings of the International Geometry Center 10 (2017) 1-21.

with Pierre Will:
A complex hyperbolic Riley slice
Geometry and Topology 21-6 (2017) 3391-3451.

with Angel Cano and José Seade:
Action of R-Fuchsian groups on CP²
The Asian Journal of Mathematics, 20 (2016) 449-474.

with Martin Deraux and Julien Paupert:
New non-arithmetic complex hyperbolic lattices
Inventiones Mathematicae, 203 (2016) 681-771.

with Jieyan Wang and Baohua Xie:
Complex hyperbolic (3,3,n) triangle groups
Pacific Journal of Mathematics, 280 (2016) 433-453.

with Krishnendu Gongopadhyay and Shiv Parsad:
On the classification of unitary matrices
Osaka Journal of Mathematics,52 (2015) 959-991.

with Pierre Will:
Complex hyperbolic free groups with many parabolic elements
Contemporary Mathematics 639 (2015) 327-348.

with Richard K Boadi:
Mostow's lattices and cone metrics on the sphere
Advances in Geometry 15 (2015) 27-53.

with Krishnendu Gongopadhyay:
Reversible complex hyperbolic isometries
Linear Algebra and its Applications 438 (2013) 2728-2739.

Traces in complex hyperbolic geometry
Geometry, Topology and Dynamics of Character Varieties,
Lecture Notes Series 23, Institute for Mathematical Sciences, National University of Singapore, 191-245.
World Scientific Publishing Co, 2012.

with Shigeyasu Kamiya and James M Thompson:
Non-discrete complex hyperbolic triangle groups of type (n,n,∞;k)
Canadian Mathematical Bulletin 55 (2012) 329-338.

with Martin Deraux and Julien Paupert:
Census of the complex hyperbolic sporadic groups
Experimental Mathematics 20 (2011) 467-486.

with Wensheng Cao:
Jørgensen's inequality and collars in n-dimensional quaternionic hyperbolic space
Quarterly Journal of Mathematics 62 (2011) 523-543.

with Elisha Falbel, Gábor Francsics and Peter D Lax:
Generators of a Picard modular group in two complex dimensions
Proceedings of the American Mathematical Society 139 (2011) 2439-2447.
with Elisha Falbel and Gábor Francsics:
The geometry of the Gauss-Picard modular group
Mathematische Annalen 349 (2011) 459-508.
with Shigeyasu Kamiya and James M Thompson:
Notes on complex hyperbolic triangle groups
Conformal Geometry and Dynamics 14 (2010) 202-218.
with Ioannis D Platis:
Complex hyperbolic quasi-Fuchsian groups
Geometry of Riemann Surfaces
London Mathematical Society Lecture Notes 368 (2010) 309-355.
Complex hyperbolic lattices
Discrete Groups and Geometric Structures
Contemporary Mathematics 501 AMS, (2009) 1-42
with Ioannis D Platis:
Global, geometrical coordinates on Falbel's cross-ratio variety
Canadian Mathematical Bulletin 52 (2009) 285-294.
with Julien Paupert:
Unfaithful complex hyperbolic triangle groups II: Higher order reflections
Pacific Journal of Mathematics 239 (2009) 357-389.
with Ian Short:
Conjugacy classification of quaternionic Möbius transformations
Computational Methods and Function Theory 9 (2009) 13-25.
Unfaithful complex hyperbolic triangle groups I: Involutions
Pacific Journal of Mathematics 238 (2008) 145-169.
with Shigeyasu Kamiya:
Discrete subgroups of PU(2,1) with screw parabolic elements
Mathematical Proceedings of the Cambridge Philosophical Society. 144 (2008) 443-455.
Hyperbolic Spaces
Jyväskylä Lectures in Mathematics 2, 2008.
with Ioannis D Platis:
Complex hyperbolic Fenchel-Nielsen coordinates
Topology 47 (2008) 101-135.
with J Vernon Armitage:
Jørgensen's inequality for non-Archimedean metric spaces
Geometry and Dynamics of Groups and Spaces
Progress in Mathematics 265 Birkhäuser, (2007) 95-109.
with Sarah Markham:
Jørgensen's inequality for metric spaces with application to the octonions
Advances in Geometry 7 (2007) 19-38.
Cone metrics on the sphere and Livné's lattices
Acta Mathematica 196 (2006) 1-64.
with Ioannis D Platis:
Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space
Journal of Differential Geometry 73 (2006) 319-350.
with Elisha Falbel:
The geometry of the Eisenstein-Picard modular group
Duke Mathematical Jornal 131 (2006) 249-289.
with Ioannis D Platis:
Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space (research announcement)
Proceedings of the 10th Panhellenic Conference on Mathematical Analysis, Athens (2004), 199--205.
with A.D.Mednykh and A.Yu.Vesnin:
On hyperbolic polyhedra arising as convex cores of quasi-Fuchsian punctured torus groups
Boletin de la Sociedad Matematica Mexicana 10 (3) (2004) 357--381.
with Cao Wensheng and Wang Xiantao:
On the classification of quaternionic Möbius transformations
Mathematical Proceedings of the Cambridge Philosophical Society 137 (2004) 349-361.
On the stable basin theorem
Canadian Mathematical Bulletin 47 (2004) 439-444.
with Caroline Series:
The mapping class group of the twice punctured torus
Groups: Topological, Combinatorial and Arithmetic Aspects Ed: T.W. Müller
London Mathematical Society Lecture Notes 311 (2004) 405-486
with Inkang Kim:
Geometry of quaternionic hyperbolic manifolds
Mathematical Proceedings of the Cambridge Philosophical Society 135 (2003) 291-320.
with Christof Menzel:
Pseudo-Anosov diffeomorphisms of the twice punctured torus
Recent Advances in Group Theory and Low-Dimensional Topology Ed: Jung Rae Cho, Jens Mennicke
Research and Exposition in Mathematics 27 (2003) 141-154.
with Jiang Yueping and Shigeyasu Kamiya:
Jørgensen's inequality for complex hyperbolic space
Geometriae Dedicata 97 (2003) 55-80
with Nikolay Gusevskii:
Complex hyperbolic quasi-Fuchsian groups and Toledo's invariant
Geometriae Dedicata 97 (2003) 151-185
with Sarah Markham:
Collars in complex and quaternionic hyperbolic manifolds
Geometriae Dedicata 97 (2003) 199-213
with Jiang Yueping:
Uniform discreteness and Heisenberg screw motions
Mathematische Zeitschrift 243 (2003) 653--669
with Elisha Falbel:
The moduli space of the modular group in complex hyperbolic geometry
Inventiones Mathematicae 152 (2003) 57-88
Tetrahedral decomposition of punctured torus bundles
Kleinian Groups and Hyperbolic 3-Manifolds Ed: Y. Komori, V. Markovic, C. Series
London Math. Soc. Lec. Notes 299 (2003) 275-291.
with Shigeyasu Kamiya:
On discrete subgroups of PU(1,2;C) with Heisenberg translations II
Revue Roumaine Math. Pures et Appl. 47 (5-6) (2002) 687-693.
with Shigeyasu Kamiya:
Notes on discrete subgroups of PU(1,2;C) with Heisenberg translations IV
Hyperbolic Spaces and Discrete Groups II Ed: S. Kamiya
RIMS Kokyuroku 1270 (2002) 138-144.
with Jiang Yueping:
Uniform discreteness and Heisenberg screw motions (research announcement)
Hyperbolic Spaces and Discrete Groups II Ed: S. Kamiya
RIMS Kokyuroku 1270 (2002) 145-152.
with Bernd O. Stratmann:
Kleinian groups with singly cusped parabolic fixed points,
Kodai Mathematical Journal 24 (2001) 169-206.
with Nikolay Gusevskii:
Representations of free Fuchsian groups in complex hyperbolic space,
Topology 39 (1) (2000) 33-60.
with Linda Keen and Caroline Series:
Combinatorics of simple closed curves on the twice punctured torus
Israel J. of Maths. 112 (1999) 29-60.
with Jouni Parkkonen:
Coordinates for quasi-Fuchsian punctured torus space,
Geometry and Topology Monographs 1 (1998), The Epstein Birthday Schrift, 451-478.
On the volumes of cusped complex hyperbolic manifolds and orbifolds,
Duke Math. J. 94 (1998) 433-464.
Uniform discreteness and Heisenberg translations,
Math. Zeit. 225 (1997) 485-505.
with Caroline Series:
Bending formulae for convex hull boundaries,
Journal d'Analyse Mathématique 67 (1995) 165-198.
Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space,
Geometriae Dedicata 57 (1995) 223-234.
Kleinian circle packings,
Topology 34 (1995) 489-496.
with Greg McShane and Ian Redfern:
Drawing limit sets of Kleinian groups using finite state automata,
Experimental Mathematics, 3 (1994) 153-170.
On Ford isometric spheres in complex hyperbolic space,
Mathematical Proceedings of the Cambridge Philosophical Society 115 (1994) 501-512.
Shimizu's lemma for complex hyperbolic space,
International J. Math. 3 (1992) 291-308.
with William M. Goldman:
Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space,
J. of Geometric Analysis, 2 (1992) 517-554.
with William M. Goldman:
Complex hyperbolic ideal triangle groups,
J. reine angewandte Math., 425 (1992) 71-86.
Rational powers of generators of Möbius groups,
Michigan Math. J., 39 (1992) 309-323.
2-generator Möbius groups,
Ph.D. thesis, University of Cambridge (1989).