Maximal subgroups of infinite index in finitely generated linear groups

Margulis, G. A.; Soifer, G. A.

doi:10.1016/0021-8693(81)90123-x

Cited by 68 publications

(58 citation statements)

References 3 publications

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“…In particular, this encompasses a result of Margulis and Soifer (the easier implication in main Theorem of [MS81]). The main ingredient to prove Proposition 3.20 is the following deep result:…”

Section: Proposition 320 Let G Be Group Which Is Nilpotent-by-(virmentioning

confidence: 92%

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Finitely Presented Wreath Products and Double Coset Decompositions

Cornulier

2006

Geom Dedicata

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Abstract. We characterize which permutational wreath products G ⋉ W (X) are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X 2 .On the one hand, this extends a result of G. Baumslag about infinite presentation of standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup.Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.

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Section: Proposition 320 Let G Be Group Which Is Nilpotent-by-(virmentioning

confidence: 92%

“…2) In [MS81], it is proved that a finitely generated group which is linear over a commutative ring, and not virtually solvable, has a maximal subgroup of infinite index, thus does not have Property (MF).…”

Section: Lemma 32 For Pairs (G H) We Have the Implications: (H Hamentioning

confidence: 99%

Finitely Presented Wreath Products and Double Coset Decompositions

Cornulier

2006

Geom Dedicata

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“…[12] Proof. If N is a non-central normal subgroup of D * , then, as we saw before, there is a finitely generated integral domain R such that N < GL n (R), where n is the dimension of D over F .…”

Section: Let Us Consider the Elementmentioning

confidence: 99%

“…Also, in Chapter 3 of [4], there are several group theoretic conditions whose occurrence in the multiplicative group of a division algebra entails the commutativity of the algebra. Much research in recent years has been focused on the study of the structure of the multiplicative subgroups of division algebras; for example see [1], [3], [6], [7], [8], [10], [11], and [12] for an introduction. Before stating our results, we fix some notation.…”

Section: Introductionmentioning

confidence: 99%

Normal subgroups of $GL_n(D)$ are not finitely generated

Akbari

Mahdavi-Hezavehi

1999

Proc. Amer. Math. Soc.

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“…[24]). In the higher rank case there are, for instance, the generalized Schottky groups constructed by Y. Benoist, the groups considered by V. G. Kac and E. B. Vinberg and those studied by G. A. Margulis and G. A. Soifer (see [3], [12] and [16] respectively).…”

Section: Introductionmentioning

confidence: 99%

Critical exponents of discrete groups and 𝐿²–spectrum

Leuzinger

2003

Proc. Amer. Math. Soc.

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Abstract. Let G be a noncompact semisimple Lie group and Γ an arbitrary discrete, torsion-free subgroup of G. Let λ 0 (M ) be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space M = Γ\X, and let δ(Γ) be the exponent of growth of Γ. If G has rank 1, then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating λ 0 (M ) from above and below by quadratic polynomials in δ(Γ). As an application we prove a rigiditiy property of lattices.

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Maximal subgroups of infinite index in finitely generated linear groups

Cited by 68 publications

References 3 publications

Finitely Presented Wreath Products and Double Coset Decompositions

Finitely Presented Wreath Products and Double Coset Decompositions

Normal subgroups of $GL_n(D)$ are not finitely generated

Critical exponents of discrete groups and 𝐿²–spectrum

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