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The congruence subgroup problem
, 2003
"... 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 nonspecialists and avoids technical details. ..."
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Cited by 32 (0 self)
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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 nonspecialists and avoids technical details.
Congruence Subgroup Property
"... Abstract. This module will explain what the Congruence Subgroup Property is, and why it is important. Then “Mennicke symbols ” (a tool from Algebraic KTheory) will be used to show that SL(3, Z) has the property, and a stronger property called “bounded generation.” ..."
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Abstract. This module will explain what the Congruence Subgroup Property is, and why it is important. Then “Mennicke symbols ” (a tool from Algebraic KTheory) will be used to show that SL(3, Z) has the property, and a stronger property called “bounded generation.”
THE CONGRUENCE SUBGROUP PROBLEM FOR UNITS
"... Let K be a number field. Denote its unit group O × K by UK. For any nonzero ideal c in OK, let UK(c) = {u ∈ UK: u ≡ 1 mod c}. This is the kernel of UK → (OK/c)×. A subgroup of UK which contains UK(c) for some c is called a congruence subgroup. For a subgroup Γ ⊂ UK which contains UK(c), any u ∈ UK ..."
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Let K be a number field. Denote its unit group O × K by UK. For any nonzero ideal c in OK, let UK(c) = {u ∈ UK: u ≡ 1 mod c}. This is the kernel of UK → (OK/c)×. A subgroup of UK which contains UK(c) for some c is called a congruence subgroup. For a subgroup Γ ⊂ UK which contains UK(c), any u ∈ UK
CONGRUENCE SUBGROUPS AND THE ATIYAH CONJECTURE
, 2005
"... Abstract. Let Q denote the algebraic closure of Q in C. Suppose G is a torsionfree group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the Calgebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove tha ..."
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Cited by 3 (1 self)
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Abstract. Let Q denote the algebraic closure of Q in C. Suppose G is a torsionfree group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the Calgebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove
THE STABLE HOMOLOGY OF CONGRUENCE SUBGROUPS
"... 0.1. Introduction. Let F be a number field, and let Γ = SLN(OF). For an integer M, let Γ(M) denote the principal congruence subgroup of level M. The cohomology of Γ in any fixed degree is well known to be stable as N → ∞ stable in fixed degree [Cha80]. The cohomology of Γ(M), however, does not stab ..."
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Cited by 4 (2 self)
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0.1. Introduction. Let F be a number field, and let Γ = SLN(OF). For an integer M, let Γ(M) denote the principal congruence subgroup of level M. The cohomology of Γ in any fixed degree is well known to be stable as N → ∞ stable in fixed degree [Cha80]. The cohomology of Γ(M), however, does
Stability in the homology of congruence subgroups
 Invent. Math
"... The homology groups of many natural sequences of groups fGng¥n=1 (e.g. general linear groups, mapping class groups, etc.) stabilize as n! ¥. Indeed, there is a wellknown machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of grou ..."
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Cited by 9 (2 self)
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of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations. They called this phenomena representation stability. We prove that the homology groups of congruence subgroups of GLn(R) (for almost any reasonable ring R) satisfy a strong version
CONGRUENCE SUBGROUPS IN THE HURWITZ QUATERNION ORDER
, 2008
"... Abstract. We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. We present some properties of the associated congruence subgroups. Namely, we show that a Hurwitz group defined by a congruence subgroup as ..."
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Abstract. We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry. We present some properties of the associated congruence subgroups. Namely, we show that a Hurwitz group defined by a congruence subgroup
Identifying congruence subgroups of the modular subgroup
 Proceedings of the American Mathematical Society 124
, 1996
"... Abstract. We exhibit a simple test (Theorem 2.4) for determining if a given (classical) modular subgroup is a congruence subgroup, and give a detailed description of its implementation (Theorem 3.1). In an appendix, we also describe a more \invariant " and arithmetic congruence test. 1. Notatio ..."
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Cited by 12 (0 self)
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Abstract. We exhibit a simple test (Theorem 2.4) for determining if a given (classical) modular subgroup is a congruence subgroup, and give a detailed description of its implementation (Theorem 3.1). In an appendix, we also describe a more \invariant " and arithmetic congruence test. 1
Results 1  10
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19,089