Almost all Bianchi groups have free, non-cyclic quotients

1992 ◽  
Vol 111 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. W. Mason ◽  
R. W. K. Odoni ◽  
W. W. Stothers
Keyword(s):  
Congruence Subgroup ◽  
Linear Groups ◽  
Discrete Groups ◽  
Quadratic Number Field ◽  
Low Dimensional Topology ◽  
Imaginary Quadratic Number ◽  
Low Dimensional ◽  
Almost All ◽  
Discontinuous Groups ◽  
Groups Of Isometries

Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.

Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices

1994 ◽  
Vol 116 (2) ◽  
pp. 253-273
Author(s):  
A. W. Mason ◽  
R. W. K. Odoni
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Quadratic Number ◽  
Congruence Subgroups ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Imaginary Quadratic Number ◽  
Low Dimensional ◽  
Fundamental Paper

AbstractLet d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

New Perspectives on the Interplay between Discrete Groups in Low-Dimensional Topology and Arithmetic Lattices

2015 ◽  
Vol 12 (2) ◽  
pp. 1701-1746
Author(s):  
Ursula Hamenstädt ◽  
Gregor Masbaum ◽  
Alan Reid ◽  
Tyakal Nanjundiah Venkataramana
Keyword(s):  
Discrete Groups ◽  
Low Dimensional Topology ◽  
Low Dimensional

scholarly journals Chebyshev polynomials and inequalities for Kleinian groups

2021 ◽  
Author(s):  
Hala Alaqad ◽  
Jianhua Gong ◽  
Gaven Martin
Keyword(s):  
Chebyshev Polynomials ◽  
Kleinian Groups ◽  
Necessary Condition ◽  
Complex Numbers ◽  
Discrete Groups ◽  
Low Dimensional Topology ◽  
Principal Character ◽  
Group A ◽  
Low Dimensional ◽  
Jørgensen’S Inequality

The principal character of a representation of the free group of rank two into [Formula: see text] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

Low-Dimensional Topology and Number Theory

2010 ◽  
pp. 2101-2163 ◽  
Author(s):  
Paul Gunnells ◽  
Walter Neumann ◽  
Adam Sikora ◽  
Don Zagier
Keyword(s):  
Number Theory ◽  
Low Dimensional Topology ◽  
Low Dimensional

scholarly journals Cross-sections of unknotted ribbon disks and algebraic curves

2019 ◽  
Vol 155 (2) ◽  
pp. 413-423
Author(s):  
Kyle Hayden
Keyword(s):  
Cross Sections ◽  
Algebraic Curves ◽  
American Mathematical Society ◽  
Geometric Topology ◽  
Mathematical Society ◽  
Advanced Mathematics ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Holomorphic Disks

We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

scholarly journals Algebraic Structures in Low-Dimensional Topology

2014 ◽  
Vol 11 (2) ◽  
pp. 1403-1458 ◽  
Author(s):  
Louis Kauffman ◽  
Vassily Manturov ◽  
Kent Orr ◽  
Robert Schneiderman
Keyword(s):  
Algebraic Structures ◽  
Low Dimensional Topology ◽  
Low Dimensional
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