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Modular group: History, Congruence subgroups & Relationship to hyperbolic geometry

In mathematics, the modular group is the projective special linear group PSL ⁡ ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2 {\displaystyle 2\times 2} matrices with integer coefficients and determinant 1 {\displaystyle 1} , such that the matrices A {\displaystyle A} and − A {\displaystyle -A} are identified. The modular group…

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Modular group topic overview

The analysis highlights History, Congruence subgroups and Relationship to hyperbolic geometry as prominent areas in the source structure around Modular group.

Related topics
101
Source areas
10
Connected nodes
111
Extracted relationships
51
Concept neighborhoods
52
Bridge connections
111

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Congruence subgroups · 20 topics
Relationship to hyperbolic geometry · 17 topics
Group-theoretic properties · 14 topics
Number-theoretic properties · 13 topics
Definition · 9 topics
Overview · 9 topics
Dyadic monoid · 8 topics
History · 6 topics
Maps of the torus · 4 topics
Hecke groups · 1 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Definition

Number-theoretic properties

Group-theoretic properties

Relationship to hyperbolic geometry

Congruence subgroups

Dyadic monoid

Maps of the torus

Hecke groups

History

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Modular group connects Entity context

The extracted context around Modular group shows recurring relationship patterns in the source. For example, Modular group → Important, PSL, SL, The, There, This, We, Z/NZ Another extracted example is Modular group → Cantor, Koch, Minkowski's, One, Rham, STn1STn2STn3, The, This. Use these groups to spot repeated connection types before inspecting the individual relationships.

Modular group

Top relations

related to Congruence subgroups · 8
Modular group → Important, PSL, SL, The, There, This, We, Z/NZ
related to Dyadic monoid · 8
Modular group → Cantor, Koch, Minkowski's, One, Rham, STn1STn2STn3, The, This
related to history · 8
Modular group → Carl Gustav Jakob Jacobi, Erlangen, Felix Klein, However, Joseph Louis Lagrange, Niels Henrik Abel, Richard Dedekind, The
related to Braid group · 5
Modular group → B3, Further, PSL2, SL2, The
related to Hecke groups · 5
Modular group → Erich Hecke, Hecke, Hq, The, The Hecke
related to Relationship to hyperbolic geometry · 5
Modular group → If, In, Möbius, PSL, The
related to Tessellation of the hyperbolic plane · 5
Modular group → Care, PSL, The, There, This
is a · 4
Modular group → dyadic monoid, projective special linear group PSL, so-called moduli space of elliptic curves, subgroup of the group of orientation-preserving isometries of H.Tessellation of the hyperbolic planeThe modular group Γ acts on H
related to Presentation · 2
Modular group → Geometrically, The
related to Definition · 1
Modular group → The

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

group modular displaystyle subgroup mathbb psl operatorname subgroups groups plane matrices sl triangle transformations matrix hyperbolic congruence upper half-plane linear

Modular group relationships Subject–Predicate–Object triples

TTTA extracted 51 structured relationships around Modular group. Examples in this analysis include Modular group → is a → projective special linear group PSL and Modular group → is a → subgroup of the group of orientation-preserving isometries of H.Tessellation of the hyperbolic planeThe modular group Γ acts on H. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Modular groupis aprojective special linear group PSL0.90text
Modular groupis asubgroup of the group of orientation-preserving isometries of H.Tessellation of the hyperbolic planeThe modular group Γ acts on H0.90text
Modular groupis aso-called moduli space of elliptic curves0.90text
Modular groupis adyadic monoid0.90text
Modular grouprelated to Braid groupThe0.60section
Modular grouprelated to Braid groupB30.60section
Modular grouprelated to Braid groupSL20.60section
Modular grouprelated to Braid groupPSL20.60section
Modular grouprelated to Braid groupFurther0.60section
Modular grouprelated to Congruence subgroupsImportant0.60section
Modular grouprelated to Congruence subgroupsThere0.60section
Modular grouprelated to Congruence subgroupsSL0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Modular group bring nearby vocabulary together. In this analysis, examples include Modular, Groups and Subgroup. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Modular group
    • Modular
    • Groups
    • Subgroup
    • Plane
    • Subgroups
    • Transformations
    • Congruence
    • Half-plane
    • Hyperbolic
    • Upper
    • Sl
    • Psl
  • modular group
    • Modular
    • Groups
    • Subgroup
    • Psl
    • Plane
    • Subgroups
    • Transformations
    • Half-plane
    • Mathbb
    • Displaystyle
    • Form
    • Congruence
  • projective special linear group
    • Modular
    • Transformations
    • Monoid
    • Also
    • Form
    • Displaystyle
    • Groups
    • Plane
    • Psl
    • Sl
    • Operatorname
    • Half-plane
  • linear fractional transformations
    • Transformations
    • Monoid
    • Upper
    • Also
    • Form
    • Displaystyle
    • Plane
    • Sl
    • Operatorname
    • Psl
    • Mathbb
    • Gl
  • modular arithmetic
    • Groups
    • Subgroup
    • Plane
    • Subgroups
    • Transformations
    • Congruence
    • Half-plane
    • Hyperbolic
    • Upper
    • Psl
    • Elliptic
    • Form
  • group
    • Modular
    • Groups
    • Psl
    • Half-plane
    • Plane
    • Transformations
    • Subgroup
    • Mathbb
    • Displaystyle
    • Form
    • Isomorphic
    • Hyperbolic
  • fractional linear transformations
    • Transformations
    • Monoid
    • Upper
    • Also
    • Form
    • Displaystyle
    • Plane
    • Sl
    • Operatorname
    • Psl
    • Mathbb
    • Gl
  • complex upper half-plane
    • Half-plane
    • Upper
    • Elliptic
    • Hyperbolic
    • Plane
    • Transformations
    • Quotient
    • Curve
    • Form
    • Numbers
    • Congruence
    • Subgroups

Connections between topic areas Semantic bridges

For Modular group, one of the stronger structural bridges in this analysis connects Modular group with Congruence subgroups. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Modular group — Congruence subgroups · splits 91 ⟂ 21
Modular group — Relationship to hyperbolic geometry · splits 94 ⟂ 18
Modular group — Group-theoretic properties · splits 97 ⟂ 15
Modular group — Number-theoretic properties · splits 98 ⟂ 14
Modular group — Overview · splits 102 ⟂ 10
Modular group — Definition · splits 102 ⟂ 10
Modular group — Dyadic monoid · splits 103 ⟂ 9
Modular group — History · splits 105 ⟂ 7
Modular group — Maps of the torus · splits 107 ⟂ 5

Map overview Semantic statistics

Modular group

Nodes112
Edges111
Triples51
Avg. degree1.98
Density0.017857
Components1

Source & methodology

TTTA analyzes the structure around Modular group to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as History, Congruence subgroups & Relationship to hyperbolic geometry, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Modular group · EN edition · Analysis: TopicsToTalkAbout

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