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
 email:

j.r.parker@durham.ac.uk

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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
JeanMarc Schlenker and I were joint EditorsinChief of
Geometriae Dedicata. From January 2023
Richard Wentworth has taken over from us as EditorinChief.
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.
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) 380389.

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) 37813796.

with
Martin Deraux
and
Julien Paupert:
New nonarithmetic complex hyperbolic lattices II
Martin Deraux's software used in this project
spocheck.
Michigan Mathematical Journal
70 (2021) 133205.

with Hirotaka Akiyoshi,
Ken'ichi Ohshika,
Makoto Sakuma
and Han Yoshida:
Classification of nonfree Kleinian groups generated by two parabolic transformations
Transactions of the American Mathematical Society
374 (2021) 17651814.

Nonarithmetic monodromy of higher hypergeometric functions
Journal d'Analyse Mathématique
142 (2020) 4170.

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

with Andrew Monaghan and
and
Anna Pratoussevitch:
Discreteness of ultraparallel complex hyperbolic triangle groups of
type [m_{1},m_{2},0]
Journal of the London Mathematical Society
100 (2019) 545567.

with
Norbert Peyerimhoff and
Karl Friedrich Siburg:
Minimizing length of billiard trajectories in hyperbolic polygons
Conformal Geometry and Dynamics
22 (2018) 315332.

with Wensheng Cao:
Shimizu's lemma for quaternionic hyperbolic space
Computational Methods and Function Theory
18 (2018) 159191.

On cusp regions associated to screwparabolic maps
Geometriae Dedicata 192 (2018) 267294.

with Lijie Sun:
Complex hyperbolic triangle groups with 2fold symmetry
Proceedings of the International Geometry Center
10 (2017) 121.

with Pierre Will:
A complex hyperbolic Riley slice
Geometry and Topology 216 (2017) 33913451.

with Angel Cano
and José Seade:
Action of RFuchsian groups on CP^{2}
The Asian Journal of Mathematics, 20 (2016) 449474.

with
Martin Deraux
and
Julien Paupert:
New nonarithmetic complex hyperbolic lattices
Inventiones Mathematicae,
203 (2016) 681771.

with Jieyan Wang and Baohua Xie:
Complex hyperbolic (3,3,n) triangle groups
Pacific Journal of Mathematics,
280 (2016) 433453.

with Krishnendu
Gongopadhyay and Shiv Parsad:
On the classification of unitary matrices
Osaka Journal of Mathematics,52 (2015) 959991.

with Pierre Will:
Complex hyperbolic free groups with many parabolic elements
Contemporary Mathematics 639 (2015) 327348.

with Richard K Boadi:
Mostow's lattices and cone metrics on the sphere
Advances in Geometry 15 (2015) 2753.

with Krishnendu
Gongopadhyay:
Reversible complex hyperbolic isometries
Linear Algebra and its Applications 438 (2013) 27282739.

Traces in complex hyperbolic geometry
Geometry, Topology and Dynamics of Character Varieties,
Lecture Notes Series 23, Institute for Mathematical
Sciences, National University of Singapore, 191245.
World Scientific Publishing Co, 2012.

with
Shigeyasu Kamiya and James M Thompson:
Nondiscrete complex hyperbolic triangle groups of type (n,n,∞;k)
Canadian Mathematical Bulletin 55 (2012) 329338.

with
Martin Deraux
and
Julien Paupert:
Census of the complex hyperbolic sporadic groups
Experimental Mathematics 20 (2011) 467486.

with Wensheng Cao:
Jørgensen's inequality and collars in ndimensional quaternionic
hyperbolic space
Quarterly Journal of Mathematics 62 (2011) 523543.

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) 24392447.

with
Elisha Falbel and
Gábor Francsics:
The geometry of the GaussPicard modular group
Mathematische Annalen 349 (2011) 459508.

with
Shigeyasu Kamiya and James M Thompson:
Notes on complex hyperbolic triangle groups
Conformal Geometry and Dynamics 14 (2010) 202218.

with
Ioannis D Platis:
Complex hyperbolic quasiFuchsian groups
Geometry of Riemann Surfaces
London Mathematical Society Lecture Notes 368 (2010) 309355.

Complex hyperbolic lattices
Discrete Groups and Geometric Structures
Contemporary Mathematics 501 AMS, (2009) 142

with
Ioannis D Platis:
Global, geometrical coordinates on Falbel's crossratio variety
Canadian Mathematical Bulletin 52 (2009) 285294.

with
Julien Paupert:
Unfaithful complex hyperbolic triangle groups II: Higher order reflections
Pacific Journal of Mathematics 239 (2009) 357389.

with Ian Short:
Conjugacy classification of quaternionic Möbius transformations
Computational Methods and Function Theory 9 (2009) 1325.

Unfaithful complex hyperbolic triangle groups I: Involutions
Pacific Journal of Mathematics 238 (2008) 145169.

with
Shigeyasu Kamiya:
Discrete subgroups of PU(2,1) with screw parabolic elements
Mathematical Proceedings of the Cambridge Philosophical Society.
144 (2008) 443455.

Hyperbolic Spaces
Jyväskylä Lectures in Mathematics 2, 2008.

with
Ioannis D Platis:
Complex hyperbolic FenchelNielsen coordinates
Topology 47 (2008) 101135.

with J Vernon Armitage:
Jørgensen's inequality for nonArchimedean metric spaces
Geometry and Dynamics of Groups and Spaces
Progress in Mathematics 265 Birkhäuser, (2007) 95109.

with Sarah Markham:
Jørgensen's inequality for metric spaces with application to the
octonions
Advances in Geometry 7 (2007) 1938.

Cone metrics on the sphere and Livné's lattices
Acta Mathematica
196 (2006) 164.

with
Ioannis D Platis:
Open sets of maximal dimension in complex hyperbolic quasiFuchsian space
Journal of Differential
Geometry 73 (2006) 319350.

with
Elisha Falbel:
The geometry of the EisensteinPicard modular group
Duke Mathematical Jornal 131 (2006) 249289.

with
Ioannis D Platis:
Open sets of maximal dimension in complex hyperbolic quasiFuchsian space
(research announcement)
Proceedings of the 10th Panhellenic Conference on Mathematical Analysis,
Athens (2004), 199205.

with A.D.Mednykh and A.Yu.Vesnin:
On hyperbolic polyhedra arising as convex cores of quasiFuchsian
punctured torus groups
Boletin de la Sociedad Matematica Mexicana 10 (3) (2004)
357381.

with Cao Wensheng and Wang Xiantao:
On the classification of quaternionic Möbius transformations
Mathematical Proceedings of the Cambridge Philosophical Society
137 (2004) 349361.

On the stable basin theorem
Canadian Mathematical Bulletin 47 (2004) 439444.

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) 405486

with Inkang Kim:
Geometry of quaternionic hyperbolic manifolds
Mathematical Proceedings of the Cambridge Philosophical Society
135 (2003) 291320.

with
Christof Menzel:
PseudoAnosov diffeomorphisms of the twice punctured torus
Recent Advances in Group Theory and LowDimensional Topology
Ed: Jung Rae Cho, Jens Mennicke
Research and Exposition in Mathematics 27 (2003) 141154.

with Jiang Yueping and
Shigeyasu Kamiya:
Jørgensen's inequality for complex hyperbolic space
Geometriae Dedicata 97 (2003) 5580

with Nikolay Gusevskii:
Complex hyperbolic quasiFuchsian groups and Toledo's invariant
Geometriae Dedicata 97 (2003) 151185

with Sarah Markham:
Collars in complex and quaternionic hyperbolic manifolds
Geometriae Dedicata 97 (2003) 199213

with Jiang Yueping:
Uniform discreteness and Heisenberg screw motions
Mathematische Zeitschrift 243 (2003) 653669

with
Elisha Falbel:
The moduli space of the modular group in complex hyperbolic geometry
Inventiones Mathematicae 152 (2003) 5788

Tetrahedral decomposition of punctured torus bundles
Kleinian Groups and Hyperbolic 3Manifolds Ed: Y. Komori,
V. Markovic, C. Series
London Math. Soc. Lec. Notes 299 (2003) 275291.

with
Shigeyasu Kamiya:
On discrete subgroups of PU(1,2;C) with Heisenberg
translations II
Revue Roumaine Math. Pures et Appl. 47 (56) (2002) 687693.

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) 138144.

with Jiang Yueping:
Uniform discreteness and Heisenberg screw motions (research announcement)
Hyperbolic Spaces and Discrete Groups II Ed: S. Kamiya
RIMS Kokyuroku 1270 (2002) 145152.

with
Bernd O. Stratmann:
Kleinian groups with singly cusped parabolic fixed points,
Kodai Mathematical Journal 24 (2001) 169206.

with Nikolay Gusevskii:
Representations of free Fuchsian groups in complex hyperbolic space,
Topology
39 (1) (2000) 3360.

with Linda Keen and
Caroline Series:
Combinatorics of simple closed curves on the twice punctured torus
Israel J. of Maths. 112 (1999) 2960.

with
Jouni Parkkonen:
Coordinates for quasiFuchsian punctured torus space,
Geometry and Topology Monographs 1 (1998), The Epstein Birthday
Schrift, 451478.

On the volumes of cusped complex hyperbolic manifolds and orbifolds,
Duke Math. J. 94 (1998) 433464.

Uniform discreteness and Heisenberg translations,
Math. Zeit. 225 (1997) 485505.

with
Caroline Series:
Bending formulae for convex hull boundaries,
Journal d'Analyse Mathématique 67 (1995) 165198.

Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space,
Geometriae Dedicata 57 (1995) 223234.

Kleinian circle packings,
Topology 34 (1995) 489496.

with Greg McShane and Ian Redfern:
Drawing limit sets of Kleinian groups using finite state automata,
Experimental Mathematics, 3 (1994) 153170.

On Ford isometric spheres in complex hyperbolic space,
Mathematical Proceedings of the Cambridge Philosophical Society
115 (1994) 501512.

Shimizu's lemma for complex hyperbolic space,
International J. Math. 3 (1992) 291308.

with William M. Goldman:
Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space,
J. of Geometric Analysis, 2 (1992) 517554.

with William M. Goldman:
Complex hyperbolic ideal triangle groups,
J. reine angewandte Math., 425 (1992) 7186.

Rational powers of generators of Möbius groups,
Michigan Math. J., 39 (1992) 309323.

2generator Möbius groups,
Ph.D. thesis, University of Cambridge (1989).
Links:

Lattices in negative curvature
Borel Seminar, Les Diablerets, 17th to 21st July 2023.

Geometry of Riemann Surfaces
Madrid, 24th to 26th May 2023.

Geometry in low dimensions
Waseda University and Tokyo Tech, 27th and 28th December 2022.

Foliated
spaces, tilings and group actions
Ritsumeikan University, Japan, 17th and 18th December 2022.

Complex hyperbolic geometry and related topics
CIRM, Luminy, 4th to 8th July 2022

School
and Workshop on Geometry of Discrete Actions
ICTP, Trieste, 24th August to 4th September 2020 CANCELLED

Surfaces quotient de la boule unité et réseaux de PU(2;1)
CIRM, Luminy, 25th February to 1st March, 2019

Symposium on Geometry, Topology and Dynamics in Low Dimensions
Mathematics Institute, Warwick, 20172018
Representation varieties and geometric structures in low dimensions
2nd to 6th July 2018

Geometric Structures and Representation Varieties
Heidelberg, 19th to 22nd February 2018

Surface Group Representations and Geometric Structures
ICTS, Bangalore, 27th November to 30th November 2017

Thermodynamic formalism  Applications to geometry and number theory
In memoriam: Bernd Stratmann
Bremen University, 10th to 12th July 2017

Nonpositive curvature: group actions and cohomology
Newton Institute, Cambridge, January to June 2017

20 years of the Cuernavaca
Mathematics Institute
Cuernavaca, Mexico, 9th to 13th January 2017
My participation funded by a UK Mexico Visiting Chair.

DurhamHiroshima
Student MiniWorkshop
Hiroshima University, 2nd December 2016

Geometry, Topology and Dynamics of Moduli Spaces
National University of Singapore, 119 August 2016

Geometries, Surfaces and
Representations of Fundamental Groups
In honour of Bill Goldman's
60th birthday.
University of Maryland, 22nd to 24th June 2016

Geometric Analysis on Discrete Groups
RIMS, Kyoto University, 30th May to 3rd June 2016

Journées de Géométrie Hyperbolique
Fribourg, 15th to 16th October 2015

School
and Workshop on Geometry of Discrete Actions
ICTP, Trieste, 24th August to 4th September 2015

Modern aspects of complex geometry: conference in honour of David Minda
Cincinnati 14th to 17th May 2015

Complex Hyperbolic Geometry and Related Topics
Okayama University of Science 911 January 2015

Geometry, Arithmetic and Analyis on Hyperbolic Spaces

Borel Seminar 2014, Discrete group actions in geometry and topology
Les Diablerets, June/July 2014

GEAR Junior Retreat
University of Michigan, May/June 2014

Exotic Geometric Structures
Brown University, September 2013