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Professor John Parker's Outputs (62)

Free groups generated by two parabolic maps (2022)
Journal Article
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

In this paper we consider a group generated by two unipotent parabolic elements of SU(2, 1) with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite R-circ... Read More about Free groups generated by two parabolic maps.

Chaotic Delone Sets (2021)
Journal Article
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

We present a definition of chaotic Delone set and establish the genericity of chaos in the space of (ϵ,δ)-Delone sets for ϵ≥δ. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets... Read More about Chaotic Delone Sets.

Classification of non-free Kleinian groups generated by two parabolic transformations (2021)
Journal Article
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

We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.

New non-arithmetic complex hyperbolic lattices II (2020)
Journal Article
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

We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups. This allows us to produce new nonarithmetic lattices, bringing the number of known nonarithmetic commensurability classes in PU(2,1) to 22.

Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces (2020)
Journal Article
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

In this article we consider solvable hypersurfaces of the form with induced metrics in the symmetric space , where H a suitable unit length vector in the subgroup A of the Iwasawa decomposition . Since M is rank 2, A is 2-dimensional and we can param... Read More about Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces.

Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type [m_1,m_2,0] (2019)
Journal Article
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

In this paper, we consider ultra‐parallel complex hyperbolic triangle groups of type [m1,m2,0] , that is, groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra‐parallel complex geodesics two of which in... Read More about Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type [m_1,m_2,0].

Minimizing length of billiard trajectories in hyperbolic polygons (2018)
Journal Article
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

Closed billiard trajectories in a polygon in the hyperbolic plane can be coded by the order in which they hit the sides of the polygon. In this paper, we consider the average length of cyclically related closed billiard trajectories in ideal hyperbol... Read More about Minimizing length of billiard trajectories in hyperbolic polygons.

Shimizu’s Lemma for Quaternionic Hyperbolic Space (2017)
Journal Article
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

We give a generalisation of Shimizu’s lemma to complex or quaternionic hyperbolic space in any dimension for groups of isometries containing an arbitrary parabolic map. This completes a project begun by Kamiya (Hiroshima Math J 13:501–506, 1983). It... Read More about Shimizu’s Lemma for Quaternionic Hyperbolic Space.

A complex hyperbolic Riley slice (2017)
Journal Article
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

We study subgroups of PU(2,1) generated by two non-commuting unipotent maps A and B whose product AB is also unipotent. We call U the set of conjugacy classes of such groups. We provide a set of coordinates on U that make it homeomorphic to R2 . By c... Read More about A complex hyperbolic Riley slice.

Complex Hyperbolic Triangle Groups with 2-fold Symmetry (2017)
Journal Article
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

In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2π/p with p > 2. We will mainly concentrate on the groups where some elements are elliptic of finite order. Th... Read More about Complex Hyperbolic Triangle Groups with 2-fold Symmetry.

On cusp regions associated to screw-parabolic maps (2017)
Journal Article
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

Erlandsson and Zakeri gave a very precise description of the Margulis region associated to cusps of hyperbolic 4-manifolds associated to screw-parabolic maps. We give some bounds on the asymptotic shape of these regions that improve on the results of... Read More about On cusp regions associated to screw-parabolic maps.

Action of R-Fuchsian groups on CP2 (2016)
Journal Article
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

We look at lattices in Iso+(H2R)Iso+(HR2), the group of orientation preserving isometries of the real hyperbolic plane. We study their geometry and dynamics when they act on CP2CP2 via the natural embedding of SO+(2,1)↪SU(2,1)⊂SL(3,C)SO+(2,1)↪SU(2,1)... Read More about Action of R-Fuchsian groups on CP2.

Complex hyperbolic (3,3,n)-triangle groups (2016)
Journal Article
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

Let p,q,rp,q,r be positive integers. Complex hyperbolic (p,q,r)(p,q,r) triangle groups are representations of the hyperbolic (p,q,r)(p,q,r) reflection triangle group to the holomorphic isometry group of complex hyperbolic space H2CHℂ2, where the gene... Read More about Complex hyperbolic (3,3,n)-triangle groups.

On the classification of unitary matrices (2015)
Journal Article
Gongopadhyay, K., Parker, J. R., & Parsad, S. (2015). On the classification of unitary matrices. Osaka Journal of Mathematics, 52(4), 959-993

We classify the dynamical action of matrices in SU(p,q) using the coefficients of their characteristic polynomial. This generalises earlier work of Goldman for SU(2,1) and the classical result for SU(1,1), which is conjugate to SL(2,R). As geometrica... Read More about On the classification of unitary matrices.

Complex hyperbolic free groups with many parabolic elements (2015)
Presentation / Conference Contribution
Parker, J. R., & Will, P. (2015). Complex hyperbolic free groups with many parabolic elements. In Geometry, groups and dynamics : ICTS program : groups, geometry and dynamics, December 3-16, 2012, Almora, India (327-348). https://doi.org/10.1090/conm/639/12782

We consider in this work representations of the of the fundamental group of the 3-punctured sphere in PU(2,1) such that the boundary loops are mapped to PU(2,1) . We provide a system of coordinates on the corresponding representation variety, and ana... Read More about Complex hyperbolic free groups with many parabolic elements.

New non-arithmetic complex hyperbolic lattices (2015)
Journal Article
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

We produce a family of new, non-arithmetic lattices in TeX. All previously known examples were commensurable with lattices constructed by Picard, Mostow, and Deligne–Mostow, and fell into nine commensurability classes. Our groups produce five new dis... Read More about New non-arithmetic complex hyperbolic lattices.

Mostow's lattices and cone metrics on the sphere (2015)
Journal Article
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

In his seminal paper of 1980, Mostow constructed a family of lattices in PU(2, 1), the holomorphic isometry group of complex hyperbolic 2-space. In this paper, we use a description of these lattices given by Thurston in terms of cone metrics on the s... Read More about Mostow's lattices and cone metrics on the sphere.

Reversible complex hyperbolic isometries (2013)
Journal Article
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

Let PU(n,1) denote the isometry group of the n-dimensional complex hyperbolic space hn. An isometry g is called reversible if g is conjugate to g-1 in PU(n,1). If g can be expressed as a product of two involutions, it is called strongly reversible. W... Read More about Reversible complex hyperbolic isometries.

Traces in complex hyperbolic geometry (2012)
Presentation / Conference Contribution
Parker, J. R. (2012). Traces in complex hyperbolic geometry. In W. M. Goldman, C. Series, & S. P. Tan (Eds.), Geometry, topology and dynamics of character varieties (191-245). https://doi.org/10.1142/9789814401364_0006

We discuss the relationship between the geometry of complex hyperbolic manifolds and orbifolds and the traces of elements of the corresponding subgroup of SU(2, 1). We begin by showing how geometrical information about individual isometries is encode... Read More about Traces in complex hyperbolic geometry.

Non-Discrete Complex Hyperbolic Triangle Groups of Type (n, n, ∞; k) (2012)
Journal Article
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

A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle group... Read More about Non-Discrete Complex Hyperbolic Triangle Groups of Type (n, n, ∞; k).

Census of the complex hyperbolic sporadic triangle groups (2011)
Journal Article
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 goal of this paper is to give a conjectural census of complex hyperbolic sporadic triangle groups. We prove that only finitely many of these sporadic groups are lattices. We also give a conjectural list of all lattices among sporadic groups, and... Read More about Census of the complex hyperbolic sporadic triangle groups.

The geometry of the Gauss-Picard modular group (2011)
Journal Article
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

We give a construction of a fundamental domain for PU(2,1,\mathbbZ [i])PU(21Z[i]), that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers \mathbbZ [i]Z[i]. We obtain from that constr... Read More about The geometry of the Gauss-Picard modular group.

Generators of a Picard modular group in two complex dimensions (2011)
Journal Article
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

The goal of the article is to prove that four explicitly given transformations, two Heisenberg translations, a rotation and an involution generate the Picard modular group with Gaussian integers acting on the two dimensional complex hyperbolic space.... Read More about Generators of a Picard modular group in two complex dimensions.

Complex hyperbolic quasi-Fuchsian groups (2010)
Presentation / Conference Contribution
Parker, J. R., & Platis, I. D. (2010). Complex hyperbolic quasi-Fuchsian groups. In F. P. Gardiner, G. González-Diez, & C. Kourouniotis (Eds.), Geometry of Riemann surfaces : proceedings of the Anogia conference to celebrate the 65th birthday of William J. Harvey (309-355)

Notes on complex hyperbolic triangle groups (2010)
Journal Article
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

We first demonstrate a family of isomorphisms between complex hyperbolic triangle groups and outline a systematic approach classifying the groups. Then we describe conditions that determine the discreteness of certain groups, in particular we prove a... Read More about Notes on complex hyperbolic triangle groups.

Complex hyperbolic lattices (2009)
Presentation / Conference Contribution
Parker, J. R. (2009). Complex hyperbolic lattices. In K. Dekimpe, P. Igodt, & A. Valette (Eds.), Discrete groups and geometric structures: Workshop on Discrete Groups and Geometric Structures, with Applications III, May 26-30, 2008, Kortrijk, Belgium (1-42)

Unfaithful complex hyperbolic triangle groups II: Higher order reflections (2009)
Journal Article
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

We consider symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2π ∕ p, with p ≥ 3. We restrict our attention to those groups where certain words are elliptic. Our goal is to find necessary conditions for... Read More about Unfaithful complex hyperbolic triangle groups II: Higher order reflections.

Global, geometrical coordinates on Falbel's cross-ratio variety (2009)
Journal Article
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

Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic 2-space are completely determined, up to conjugation in PU(2,1), by three complex cross-ratios satisfying two real equations. We give global geometrical coord... Read More about Global, geometrical coordinates on Falbel's cross-ratio variety.

Conjugacy classification of quaternionic Möbius transformations (2009)
Journal Article
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

It is well known that the dynamics and conjugacy class of a complex Möbius transformation can be determined from a simple rational function of the coefficients of the transformation. We study the group of quaternionic Möbius transformations and ident... Read More about Conjugacy classification of quaternionic Möbius transformations.

Unfaithful complex hyperbolic triangle groups I: Involutions (2008)
Journal Article
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

A complex hyperbolic triangle group is the group of complex hyperbolic isometries generated by complex involutions fixing three complex lines in complex hyperbolic space. Such a group is called equilateral if there is an isometry of order three that... Read More about Unfaithful complex hyperbolic triangle groups I: Involutions.

Discrete subgroups of PU(2,1) with screw parabolic elements (2008)
Journal Article
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

We give a version of Shimizu's lemma for groups of complex hyperbolic isometries one of whose generators is a parabolic screw motion. Suppose that G is a discrete group containing a parabolic screw motion A and let B be any element of G not fixing th... Read More about Discrete subgroups of PU(2,1) with screw parabolic elements.

Complex hyperbolic Fenchel-Nielsen coordinates (2008)
Journal Article
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

Let Σ be a closed, orientable surface of genus g. It is known that the representation variety of π1(Σ) has 2g−3 components of (real) dimension 16g−16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that... Read More about Complex hyperbolic Fenchel-Nielsen coordinates.

Jorgensen's inequality for non-Archimedean metric spaces (2008)
Presentation / Conference Contribution
Armitage, J., & Parker, J. R. (2008). Jorgensen's inequality for non-Archimedean metric spaces. In M. Kapranov, S. Kolyada, Y. Manin, P. Moree, & L. Potyagailo (Eds.), Geometry and dynamics of groups and spaces : in memory of Alexander Reznikov (97-111). https://doi.org/10.1007/978-3-7643-8608-5_2

Jørgensen’s inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete. In this paper we generalise this to the case of groups of Möbius transformations of a non-Archimedean metric space. As an applicat... Read More about Jorgensen's inequality for non-Archimedean metric spaces.

Cone metrics on the sphere and Livne's lattices (2006)
Journal Article
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

We give an explicit construction of a family of lattices in PU (1, 2) originally constructed by Livné. Following Thurston, we construct these lattices as the modular group of certain Euclidean cone metrics on the sphere. We give connections between t... Read More about Cone metrics on the sphere and Livne's lattices.

Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space (2006)
Journal Article
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

Let pi(1), be the fundamental group of a closed surface Sigma of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of pi(1) into S... Read More about Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space.

The geometry of the Eisenstein-Picard modular group (2006)
Journal Article
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

The Eisenstein-Picard modular group ${\rm PU}(2,1;\mathbb {Z}[\omega])$ is defined to be the subgroup of ${\rm PU}(2,1)$ whose entries lie in the ring $\mathbb {Z}[\omega]$, where $\omega$ is a cube root of unity. This group acts isometrically and pr... Read More about The geometry of the Eisenstein-Picard modular group.

The mapping class group of the twice punctured torus (2004)
Book Chapter
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 (2003)
Book Chapter
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 (2003)
Book Chapter
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

The moduli space of the modular group in complex hyperbolic geometry (2003)
Journal Article
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

We construct the space of discrete, faithful, type-preserving representations of the modular group into the isometry group of complex hyperbolic 2-space up to conjugacy. This is the first Fuchsian group for which the entire complex hyperbolic deforma... Read More about The moduli space of the modular group in complex hyperbolic geometry.