Integrality of the group ring of an almost solvable torsion-free subgroup of GL n (Q)

1989 ◽  
Vol 45 (2) ◽  
pp. 135-140
Author(s):  
G. A. Noskov
Keyword(s):  
Group Ring ◽  
Free Subgroup ◽  
Torsion Free Subgroup ◽  
Torsion Free

The Symplectic Representation and the Torelli Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Finite Index ◽  
Algebraic Structure ◽  
Symplectic Form ◽  
Intersection Number ◽  
Free Subgroup ◽  
Torelli Group ◽  
Residually Finite ◽  
Basic Properties ◽  
Torsion Free Subgroup ◽  
Torsion Free

This chapter discusses the basic properties and applications of a symplectic representation, denoted by Ψ‎, and its kernel, called the Torelli group. After describing the algebraic intersection number as a symplectic form, the chapter presents three different proofs of the surjectivity of Ψ‎, each illustrating a different theme. It also illustrates the usefulness of the symplectic representation by two applications to understanding the algebraic structure of Mod(S). First, the chapter explains how this representation is used by Serre to prove the theorem that Mod(Sɡ) has a torsion-free subgroup of finite index. It thens uses the symplectic representation to prove, following Ivanov, the following theorem of Grossman: Mod(Sɡ) is residually finite. It also considers some of the pioneering work of Dennis Johnson on the Torelli group. In particular, a Johnson homomorphism is constructed and some of its applications are given.

A REDUCTION TO THE COMPACT CASE FOR GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

2014 ◽  
Vol 79 (01) ◽  
pp. 45-53
Author(s):  
ANNALISA CONVERSANO
Keyword(s):  
Lie Groups ◽  
Compact Subgroup ◽  
Free Subgroup ◽  
Homotopy Equivalent ◽  
Minimal Structure ◽  
Compact Case ◽  
Torsion Free Subgroup ◽  
Image Position ◽  
Torsion Free

Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M. We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K, which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$ . It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$ ), and homotopy equivalent to K. This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.

scholarly journals Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

2020 ◽  
Vol 15 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Rational Number ◽  
Prime Power ◽  
Number Fields ◽  
Free Subgroup ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Adic Valuation ◽  
Torsion Free Subgroup ◽  
Almost All ◽  
Torsion Free

AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

scholarly journals On Unit Groups of Absolute Abelian Number Fields of Degree pq

1960 ◽  
Vol 16 ◽  
pp. 73-81
Author(s):  
Hideo Yokoi
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Unit Group ◽  
Number Fields ◽  
Free Subgroup ◽  
Finite Degree ◽  
Maximal Torsion ◽  
Torsion Free Subgroup ◽  
Torsion Free ◽  
Arbitrary Abelian Group

In this note, we denote by Q the rational number field, by EΩ the whole unit group of an arbitrary number field Ω of finite degree, and by rΩ the rank of where generally G* for an arbitrary abelian group G means a maximal torsion-free subgroup of G. (NK/ΩEK)* is shortly denoted by and (G1 : G2) is, as usual, the index of a subgroup G2 in G1.

LINK COMPLEMENTS ARISING FROM ARITHMETIC GROUP ACTIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 337-370 ◽  
Author(s):  
FRITZ GRUNEWALD ◽  
ULRICH HIRSCH
Keyword(s):  
Quotient Space ◽  
Free Subgroup ◽  
Arithmetic Group ◽  
Hyperbolic Structure ◽  
3 Dimensional ◽  
Ring Of Integers ◽  
Hyperbolic Volume ◽  
Link Complements ◽  
Torsion Free Subgroup ◽  
Torsion Free

Let [Formula: see text] be a torsion-free subgroup acting discontinuously on 3-dimensional hyperbolic space [Formula: see text]. Assume further that Γ\ℍ3 has finite hyperbolic volume. The quotient-space Γ\ℍ3 is then a 3-manifold which can be compactified by the addition of finitely many 2-tori. This paper discusses a procedure which decides whether Γ\ℍ3 is homeomorphic to the complement of a link in S3. We apply our procedure to subgroups of low index in [Formula: see text], where [Formula: see text] is the ring of integers in [Formula: see text]. As a result we find new link complements having a complete hyperbolic structure coming from an arithmetic group. Finally we prove that up to conjugacy there are only finitely many commensurability classes of arithmetic subgroups [Formula: see text] so that Γ\ℍ3 is homeomorphic to the complement of a link in S3.

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

On a hyperbolic 3-manifold with some special properties

1993 ◽  
Vol 113 (1) ◽  
pp. 87-90 ◽  
Author(s):  
B. Zimmermann
Keyword(s):  
Coxeter Groups ◽  
Universal Covering ◽  
Finite Subgroup ◽  
Heegaard Splitting ◽  
Heegaard Genus ◽  
Normal Torsion ◽  
Covering Group ◽  
Minimal Index ◽  
Torsion Free Subgroup ◽  
Torsion Free

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.

On solvable centerless groups of Morley rank 3

1993 ◽  
Vol 58 (2) ◽  
pp. 546-556
Author(s):  
Mark Kelly Davis ◽  
Ali Nesin
Keyword(s):  
Nilpotent Group ◽  
General Structure ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Closed Field ◽  
Morley Rank ◽  
Nonnilpotent Group ◽  
Finite Morley Rank ◽  
Torsion Free Subgroup ◽  
Torsion Free

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

Free Subgroups of Units in Group Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 309-312 ◽  
Author(s):  
Jairo Zacarias Gonçalves
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Free Subgroup ◽  
Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Free Subgroups

AbstractIn this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.Some extensions of this results to infinite nilpotent and FC groups are also considered.

scholarly journals Hypertypes of torsion-free abelian groups of finite rank

1989 ◽  
Vol 39 (1) ◽  
pp. 21-24 ◽  
Author(s):  
H.P. Goeters ◽  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

Let G be a torsion-free abelian group of finite rank n and let F be a full free subgroup of G. Then G/F is isomorphic to T1 ⊕ … ⊕ Tn, where T1 ⊆ T2 ⊆ … ⊆ Tn ⊆ ℚ/ℤ. It is well known that type T1 = inner type G and type Tn = outer type G. In this note we give two characterisations of type Ti for 1 < i < n.

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