Congruence hulls in SLn

Mason, A. W.

doi:10.1016/0022-4049(93)90056-y

Cited by 9 publications

(4 citation statements)

References 23 publications

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“…(ii) We note that CH(S) exists by [3,Theorem 2.2]. The result now follows from (i) and [2, Theorem 3.6(iv)].…”

Section: The Modular Groupmentioning

confidence: 63%

“…In this case, CH(S) is the 'smallest' congruence subgroup containing S (see [3, p. 263] Group theoretically there appears to be little to relate a subgroup S and its congruence hull CH(S) (when it exists). It is known [3,Theorem 3.6] that if F is any finitely generated group, then there exists a subgroup T of Γ with a congruence hull such that F is isomorphic to the quotient group CH(T )/T .…”

Section: Introductionmentioning

confidence: 99%

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Infinite Simple (2, 3, n )-Groups and Congruence Hulls in the Modular Group

Mason

Pride

1999

Bulletin of the London Mathematical Society

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We prove that if n > 66 and (n, 30) = 1, then there exist uncountably many infinite simple (2, 3, n)groups, that is, groups generated by a pair of elements x, y, say, where the orders of x, y and xy are 2, 3 and n, respectively. This extends previous results of Schupp and the authors.These results are used to prove the existence of subgroups of the modular group with special arithmetic properties.

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“…(ii) We note that CH(S) exists by [3,Theorem 2.2]. The result now follows from (i) and [2, Theorem 3.6(iv)].…”

Section: The Modular Groupmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Infinite Simple (2, 3, n )-Groups and Congruence Hulls in the Modular Group

Mason

Pride

1999

Bulletin of the London Mathematical Society

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“…Lemma 3.7 is already known for the case G = SL 2 , S = {v} and Γ = SL 2 (O). See the proof of [Mas1,Theorem 3.6].…”

Section: Non-uniform Arithmetic Lattices: Discrete Resultsmentioning

confidence: 99%

“…Lemma 5.7 applies, for example, to the case G = SL 2 , S = {v} and Γ = SL 2 (O) (as demonstrated in [Za1]). It is known [Mas1,Theorem 3.1] that, when Γ = SL 2 (O), the "smallest congruence subgroup" of Γ containing U(q), q ′ ={0} U(q) • Γ(q ′ ) = Γ(q), for all q. It follows that in this case Γ(q) = U (q), for all q.…”

Section: Non-uniform Arithmetic Lattices: Profinite Resultsmentioning

confidence: 99%

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Mason

Premet²,

Sury

et al. 2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let k be a global field and let k v be the completion of k with respect to v, a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k v -rank 1. Let G = G(k v ). Let Γ be an arithmetic lattice in G and let C = C(Γ) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Γ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors isF ω , the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Γ = SL 2 (O(S)), where O(S) is the the ring of S-integers in k, with S = {v}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Γ on the Bruhat-Tits tree associated with G.

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Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

Mason

1995

Israel J. Math.

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Congruence hulls in SLn

Cited by 9 publications

References 23 publications

Infinite Simple (2, 3, n )-Groups and Congruence Hulls in the Modular Group

Infinite Simple (2, 3, n )-Groups and Congruence Hulls in the Modular Group

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

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