PhD Cambridge University
CertAdvSt Mathematics (Part III) Cambridge University
BSc Durham University
RESEARCH, TEACHING, or OTHER INTERESTS
Geometry and Topology
58
Scopus Publications
Scopus Publications
Minimising length of closed billiard trajectories on hyperbolic polygons John Parker, Manvendra Somvanshi Dynamical Systems, 2026 n a hyperbolic polygon any finite collection of closed billiard trajectories can be assigned an average length function. In this paper, we consider the average length of the collection of cyclically related closed billiard trajectories in even-sided right-angled polygons and the collection of reflectively related closed billiard trajectories in Lambert quadrilaterals with acute angle π/k. We show that in the former case the average length is minimised by the regular even- sided right-angled polygon, and in the latter case it is minimised by the Lambert quadrilateral with a reflective symmetry about its long axis. We use techniques from Teichmu ̈ller theory to prove the main theorems.
Fenchel-Nielsen coordinates for SL(3,C) representations Rodrigo Dávila Figueroa, John R. Parker Geometriae Dedicata, 2025 We define Fenchel-Nielsen coordinates for representations of surface groups to $$\\textrm{SL}(3,{\\mathbb C})$$ SL ( 3 , C ) . We also show how these coordinates relate to the classical Fenchel-Nielsen coordinates and to their generalisations by Kourouniotis, Tan, Goldman, Zhang and Parker-Platis.
Caroline Series and Hyperbolic Geometry John R. Parker, Ser Peow Tan Notices of the American Mathematical Society, 2023 Caroline Series has made important contributions to hyperbolic geometry and symbolic dynamics, influencing a generation of hyperbolic geometers and dynamicists. We give a description of some of her considerable contributions to the field.
Free groups generated by two parabolic maps Sagar B. Kalane, John R. Parker Mathematische Zeitschrift, 2023 In this paper we consider a group generated by two unipotent parabolic elements of$$\\textrm{SU}(2,1)$$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$${\\mathbb R}$$R-circle in the Heisenberg group.
Chaotic delone sets Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker Discrete and Continuous Dynamical Systems Series A, 2021 We present a definition of chaotic Delone set, and establish the genericity of chaos in the space of $(\\epsilon,\\delta)$-Delone sets for $\\epsilon\\geq \\delta$. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.
New nonarithmetic complex hyperbolic lattices II Martin Deraux, John R. Parker, Julien Paupert Michigan Mathematical Journal, 2021 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.
Classification of non-free kleinian groups generated by two parabolic transformations Hirotaka Akiyoshi, Ken’ichi Ohshika, John Parker, Makoto Sakuma, Han Yoshida Transactions of the American Mathematical Society, 2021 We give a full proof to Agol’s announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.
Discreteness of ultra-parallel complex hyperbolic triangle groups of type [m1,m2,0] Andrew Monaghan, John R. Parker, Anna Pratoussevitch Journal of the London Mathematical Society, 2019 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 intersect on the boundary. We prove some discreteness and non‐discreteness results for these groups and discuss the connection between the discreteness results and ellipticity of certain group elements.
