# The congruence subgroup problem

@article{Raghunathan2004TheCS, title={The congruence subgroup problem}, author={M. S. Raghunathan}, journal={Proceedings Mathematical Sciences}, year={2004}, volume={114}, pages={299-308} }

This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.

#### 63 Citations

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#### References

SHOWING 1-10 OF 162 REFERENCES

THE STRUCTURE OF CLASSICAL ARITHMETIC GROUPS OF RANK GREATER THAN ONE

- Mathematics
- 1973

We study the structure and describe the normal subgroups of the classical arithmetic groups of relative rank greater than one. Bibliography: 32 items.

Arithmeticity of the irreducible lattices in the semi-simple groups of rank greater than 1

- Mathematics
- 1984

ADDENDUM TO THE PAPER "THE PROBLEM OF STRONG APPROXIMATION AND THE KNESER-TITS CONJECTURE FOR ALGEBRAIC GROUPS"

- Mathematics
- 1970

This paper contains several additional observations which serve to revise and further unify the proof of the approximation theorem.

The Congruence Subgroup Problem

- Mathematics
- 1967

The results I shall describe represent work initiated by John MILNOR and myself [I], and concluded in collaboration with J.-P. Serre [2]. Serre’s discovery that our problem is very closely related to… Expand

MULTIPLICATIVE ARITHMETIC OF DIVISION ALGEBRAS OVER NUMBER FIELDS, AND THE METAPLECTIC PROBLEM

- Mathematics
- 1988

The central feature of this paper is the calculation of the metaplectic kernel for the algebraic groups determined by SL(1,D), where D is a finite-dimensional central division ring over a number… Expand

ON THE REDUCED NORM 1 GROUP OF A DIVISION ALGEBRA OVER A GLOBAL FIELD

- Mathematics
- 1992

It is proved that if the Platonov-Margulis conjecture on the standard structure of normal subgroups holds for the division algebras of index r, then it also holds for the division algebras of index… Expand

Congruence subgroup problem for anisotropic groups over semilocal rings

- Mathematics
- 1991

In Chapter I, a theorem of Margulis which gives the structure of normal subgroups ofSL(1,D) for a quaternion division algebraD over a global fieldK of characteristic not 2, is generalized to… Expand

On the congruence-subgroup problem for some anisotropic algebraic groups over number fields.

- Mathematics
- 1989

Let G be a simply connected absolutely simple algebraic group, defined over a number field k. Denote by S a finite set of places of fc, containing all the archimedean places and by 0 the ring of… Expand

On systems of generators of arithmetic subgroups of higher rank groups

- Mathematics
- 1994

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of… Expand

Arithmetic Properties of Discrete Subgroups

- Mathematics
- 1974

That the factor space of a semisimple Lie group by an arithmetic subgroup has finite volume with respect to Haar measure is well known. In this paper we study results related to the converse of this… Expand