Discreteness criteria for \(\mathcal{RP}\) groups
type: Item
instance of: scholarly article
title: Discreteness criteria for \(\mathcal{RP}\) groups
zbMATH Open document ID: 1007.20047
DOI: 10.1007/BF02785427
author: D. Kharzeev
published in: Israel Journal of Mathematics
publication date: 2002-11-10T00:00:00Z
review text: After the classification of the discrete two-generator subgroups of \(\text{PSL}(2,\mathbb{R})\) was finished [see for instance \textit{B. Fine} and \textit{G. Rosenberg}, Groups '93, Galway/St. Andrews, Vol. 1, Lond. Math. Soc. Lect. Note Ser. 211, 205-232 (1995; Zbl 0847.20044)], Gehring, Gilman and Martin introduced the class of two-generator subgroups of \(\text{PSL}(2,\mathbb{C})\) with real parameters \[ \{\langle f,g\rangle\mid\text{tr}^2f-4,\text{tr}^2g-4,\text{tr}(fgf^{-1}g^{-1})-2\in\mathbb{R}\}\subset\text{PSL}(2,\mathbb{C}). \] Here discreteness criteria are given for \(f\) hyperbolic, that is, \(\text{tr }f\in\mathbb{R}\), \(|\text{tr }f|>2\), and \(g\) elliptic of even order, that is, \(\text{tr }g\in\mathbb{R}\), \(\text{tr }g=2\cos\tfrac{r\pi}n\), \(2\leq n\) even, \(\text{gcd}(r,n)=1\).
reviewed by: Gerhard Rosenberger
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 30F40
zbMATH DE Number: 1819989
zbMATH Keywords: trace identities
zbMATH Keywords: two-generator subgroups
zbMATH Keywords: discreteness criteria
zbMATH Keywords: hyperbolic elements
zbMATH Keywords: elliptic elements
MaRDI profile type: Publication
cites work: Q3310101
cites work: Q3919948
cites work: An exposition of Poincaré's polyhedron theorem
cites work: Coxeter decompositions of hyperbolic polygons
cites work: Q4866220
cites work: Kleinian groups with real parameters
cites work: Stability and extremality in j⊘rgensen's inequality
cites work: Commutators, collars and the geometry of Möbius groups
cites work: Q4840627
cites work: Q4362859
cites work: Two-generator discrete subgroups of 𝑃𝑆𝐿(2,𝑅)
cites work: An algorithm for 2-generator Fuchsian groups
cites work: On discrete generalised triangle groups
cites work: On Discrete Groups of Mobius Transformations
cites work: A NOTE ON SUBGROUPS OF SL(2, C)
cites work: Discrete groups in three-dimensional Lobachevskij space generated by two rotations
cites work: Q3483470
cites work: Some examples of discrete groups and hyperbolic orbifolds of infinite volume
cites work: ALL DISCRETE $\mathcal{RP}$ GROUPS WHOSE GENERATORS HAVE REAL TRACES
cites work: Kleinian orbifolds uniformized by RP groups with an elliptic and a hyperbolic generator
cites work: Two-generator discrete subgroups of \(\text{Isom}(\mathbb{H}^2)\) containing orientation-reversing elements
cites work: Doubly generated Fuchsian groups
cites work: Q4039828
cites work: Some Special 2-Generator Kleinian Groups
cites work: Two parabolic generator Kleinian groups
cites work: The classification of discrete 2-generator subgroups of PSL(2,R)
cites work: All two generator Fuchsian groups
cites work: All generating pairs of all two-generator Fuchsian groups
description: scientific article; zbMATH DE number 1819989