John R. Parker's Home Page

University of Durham --- Department of Mathematical Sciences

Professor John R. Parker

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

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

Research

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.

Preprints

( --- avalable when ready). If you have Netscape you should be able to view the linked articles on screen by following the link. If this does not work...

with Shigeyasu Kamiya and James M Thompson:
Non-discrete complex hyperbolic triangle groups of type (n,n,∞;k)
To appear in Canadian Mathematical Bulletin.
with Elisha Falbel and Gábor Francsics:
The geometry of the Gauss-Picard modular group
with Wensheng Cao:
Jørgensen's inequality and collars in n-dimensional quaternionic hyperbolic space
To appear in Quarterly Journal of Mathematics.
with Elisha Falbel, Gábor Francsics and Peter D Lax:
Generators of a Picard modular group in two complex dimensions
with Shigeyasu Kamiya and James M Thompson:
Notes on complex hyperbolic triangle groups

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". If you want an offprint, let me know.

with Ioannis D Platis:
Complex hyperbolic quasi-Fuchsian groups
Geometry of Riemann Surfaces
London Mathematical Society Lecture Notes 368 309-355.
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).

Book Reviews

Review of Spherical CR Geometry and Dehn Surgery by Richard Schwartz, Annal of Maths Studies 165 (2007).
Bulletin of the American Mathematical Society 46 (2009) 369-376.
Review of The Arithmetic of Hyperbolic 3-Manifolds by Colin Maclachlan and Alan W Reid. Springer 2003.
Proceedings of the Edinburgh Mathematical Society 48 (2005) 511-512.
Review of Complex Hyperbolic Geometry by William M. Goldman, Oxford University Press 1999.
Proceedings of the Edinburgh Mathematical Society 43 (2000) 443-445.
A shorter version of this review is published in Featured Reviews, issue 2000g in Mathematical Reviews.

Teaching

See duo for details.