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

Let 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.

The Symplectic Representation and the Torelli Group

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

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.

Semiclassical limits of quantum affine spaces

Semiclassical limits of generic multi-parameter quantized coordinate rings A=$\mathcal{O}$q(kn) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsion-free subgroup of k×. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R=$\mathcal{O}$(kn), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson-prime and Poisson-primitive spectra of R onto the prime and primitive spectra of~A. The Poisson-primitive spectrum of R is then identified with the space of symplectic cores in kn in the sense of Brown and Gordon, and an example is presented (over ℂ) for which the Poisson-primitive spectrum of R is not homeomorphic to the space of symplectic leaves in kn. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.

A homotopy decomposition for the classifying space of virtually torsion-free groups and applications

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ' of Γ such that Γ' has finite cohomological dimension over . Examples of such groups include the fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.

Farrell cohomology and centralizets of elementary abelian p-subgroups

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ′ of G such that Γ′ has finite cohomological dimension over ℤ. Examples of such groups include finite groups, fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.

LINK COMPLEMENTS ARISING FROM ARITHMETIC GROUP ACTIONS

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.

On solvable centerless groups of Morley rank 3

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.

On a hyperbolic 3-manifold with some special properties

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.