Book contents
- Frontmatter
- Contents
- Introduction
- Photograph
- 1 Automorphisms of solvable groups, Part I
- 2 Automorphisms of solvable groups, Part II
- 3 A survey of groups with a single defining relation
- 4 Some algorithms for computing with finite permutation groups
- 5 Five lectures on group rings
- 6 Buildings and group amalgamations
- 7 Finite presentability of S-arithmetic groups
- 8 Efficient presentations of GL(2, ℤ) and PGL(2, ℤ)
- 9 The commutator map
- 10 Polynomial functions and representations
- 11 On questions of Brauer and Feit
- 12 The Picard group and the modular group
- 13 Factor groups of the lower central series of free products of finitely generated abelian groups
- 14 Lattice ordered groups - a very biased survey
- 15 Totally orthogonal finite groups
- 16 One-relator products of groups
- 17 The Cavicchioli groups are pairwise non-isomorphic
- 18 Congruence and non-congruence subgroups of the modular group: a survey
- 19 Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups
- 20 The Lie algebra associated to the lower central series of a group
- 21 Algebraically closed locally finite groups
- 22 On power-commutative and commutation transitive groups
- 23 Dimension function for discrete groups
- 24 Coset graphs
- 25 Nilpotent quotient algorithms
- 26 Generators of p-groups
- 27 On the matrix groups associated to the isometries of the hyperbolic plane
- 28 A characteristic subgroup of N-stable groups
- 29 The isomorphism problem for integral group rings of finite nilpotent groups
- 30 Embedding the root group geometry of 2F4(q)
- 31 On generalized Frobenius complements
- 32 Subgroups of finite index in soluble groups: I
- 33 Subgroups of finite index in soluble groups: II
- 34 Some interconnections between group theory and logic
- 35 Groups covered by abelian subgroups
- 36 Embeddings of infinite permutation groups
- 37 Maximal subgroups of sporadic groups
18 - Congruence and non-congruence subgroups of the modular group: a survey
Published online by Cambridge University Press: 05 March 2012
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Book contents
- Frontmatter
- Contents
- Introduction
- Photograph
- 1 Automorphisms of solvable groups, Part I
- 2 Automorphisms of solvable groups, Part II
- 3 A survey of groups with a single defining relation
- 4 Some algorithms for computing with finite permutation groups
- 5 Five lectures on group rings
- 6 Buildings and group amalgamations
- 7 Finite presentability of S-arithmetic groups
- 8 Efficient presentations of GL(2, ℤ) and PGL(2, ℤ)
- 9 The commutator map
- 10 Polynomial functions and representations
- 11 On questions of Brauer and Feit
- 12 The Picard group and the modular group
- 13 Factor groups of the lower central series of free products of finitely generated abelian groups
- 14 Lattice ordered groups - a very biased survey
- 15 Totally orthogonal finite groups
- 16 One-relator products of groups
- 17 The Cavicchioli groups are pairwise non-isomorphic
- 18 Congruence and non-congruence subgroups of the modular group: a survey
- 19 Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups
- 20 The Lie algebra associated to the lower central series of a group
- 21 Algebraically closed locally finite groups
- 22 On power-commutative and commutation transitive groups
- 23 Dimension function for discrete groups
- 24 Coset graphs
- 25 Nilpotent quotient algorithms
- 26 Generators of p-groups
- 27 On the matrix groups associated to the isometries of the hyperbolic plane
- 28 A characteristic subgroup of N-stable groups
- 29 The isomorphism problem for integral group rings of finite nilpotent groups
- 30 Embedding the root group geometry of 2F4(q)
- 31 On generalized Frobenius complements
- 32 Subgroups of finite index in soluble groups: I
- 33 Subgroups of finite index in soluble groups: II
- 34 Some interconnections between group theory and logic
- 35 Groups covered by abelian subgroups
- 36 Embeddings of infinite permutation groups
- 37 Maximal subgroups of sporadic groups
Summary
INTRODUCTION
My aim in this note is to give a brief survey of one particular aspect of the modular group r “ PSL2(Z), namely the balance (or rather the lack of it) between its congruence and non-congruence subgroups. Among the arithmetic subgroups (those of finite index) in r, the congruence subgroups have proved to be the most important and the most widely studied. Nevertheless, it has been known for some time that, in a certain sense,
“most of the arithmetic subgroups of Γ are
non-congruence subgroups”;
I shall describe several recent lines of investigation which, in different ways, add substance to this rather tenuous statement.
The modular group (together with the closely related groups SL2(ℤ), GL2(ℤ) and PGL2(ℤ)) is like an octopus, with tentacles reaching out into many branches of pure mathematics; a complete survey is out of the question here, but I hope that, at the very least, the bibliography will enable the reader to discover more about this fascinating group and its applications. For background reading, the classic reference is still Klein and Fricke more modern treatments of various aspects of r can be found in, Fine surveys the similarities and differences between r and the Picard group PSL2(ℤ[i]), while other classes of groups closely related to r are considered in.
ARITHMETIC SUBGROUPS
Let A denote the set of arithmetic subgroups of Γ, that is, those of finite index.
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- Proceedings of Groups - St. Andrews 1985 , pp. 223 - 234Publisher: Cambridge University PressPrint publication year: 1987
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