CSA-groups and separated free constructions

A group G is S-universal if every countable group can be embedded in a quotient of G.In this paper we give conditions under which HNN groups are S-universal.These are then applied to showing the S-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL^C)).Finally some connections are mentioned between Sg-universality and the theory of ends.Introduction.A group G is So-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are Suniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also , [11].Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9].In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3].This will be applied in 3 to showing the S?-universality of a class of Kleinian groups.Finally we make some observations on the relationship between SG;-universality and the theory of ends.1. Recall that an HNN group is a group of the form G = </" ...,t",K; rel K, t^tr1 = <P,(A)>where {L,} are subgroups of K, and {<p,} are isomorphisms L, - <p,(jL,) < K. K is called the base, {L, <p(L)} associated subgroups and </" . .., i") the free part.Groups of this type were used by Higman, Neumann and Neumann to show the S-universality of free groups of rank 2 [4].The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developedby Karrass and Solitar [5].Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an S-universal quotient is itself S-universal we getLemma.An HNN group with free part of rank > 1 is SQ-universal.

Do infinite nilpotent groups always have equipotent Abelian subgroups?

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

Existentially closed CSA-groups

Centralizers of elementary Abelian subgroups in finite p-groups

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W. We use this correspondence to find all W for which L is supersolvable. In particular, we show that the only infinite Coxeter group for which L is supersolvable is the infinite dihedral group. Also, we show how this isomorphism gives an embedding of L into the partition lattice whenever W is of type An, Bn or Dn. In addition, we give several results concerning non-broken circuit bases (NBC bases) when W is finite. We show that L is supersolvable if and only if all NBC bases are obtainable by a certain specific combinatorial procedure, and we use the lattice of parabolic subgroups to identify a natural subcollection of the collection of all NBC bases.

We prove that the Waldhausen nilpotent class group of an injective index 2\namalgamated free product is isomorphic to the Farrell-Bass nilpotent class\ngroup of a twisted polynomial extension. As an application, we show that the\nFarrell-Jones Conjecture in algebraic K-theory can be sharpened from the family\nof virtually cyclic subgroups to the family of finite-by-cyclic subgroups.\n

The purpose of this paper is to establish some basic properties of the Wielandt subgroup of a polycyclic group. The Wielandt subgroup of a group G is defined to be the intersection of the normalisers of all the subnormal subgroups of G and is denoted by ω(G). In 1958 Wielandt [9] showed that any minimal normal subgroup with the minimum condition on subnormal subgroups is contained in the Wielandt subgroup: it follows that the Wielandt subgroup of an artinian group is nontrivial. In contrast, the Wielandt subgroup of a polycyclic group can be trivial; an easy example is given by the infinite dihedral group. We will show that the Wielandt subgroup of a polycyclic group is close to being central.

Let n = 2m ≥ 4 and let q be a power of a prime p. We prove that the simple group Bn(q) is uniquely determined by the set of orders of its maximal abelian subgroups.

We conjecture the Abelian dominance in Einstein’s theory, that is, the Abelian part of the theory plays the central role in the dynamics. Treating Einstein’s theory as a gauge theory of the Lorentz group, we show that Einstein’s theory can be decomposed into the restricted part made up of the restricted connection which has the full Lorentz gauge invariance and the valence part made up of the valence connection which plays the role of gravitational source of the restricted gravity. In this decomposition, the role of the metric gμν is replaced by a four-index metric tensor gμν which transforms covariantly under the Lorentz group, and the metric-compatibility condition ∇αgμν = 0 of the connection is replaced by the gauge and generally covariant condition . We show that there are two different Abelian decompositions, the light-like (or null) decomposition and the non-light-like (or non-null) decomposition, because the Lorentz group has two maximal Abelian subgroups. The decomposition shows the existence of the restricted gravity which has the full general invariance but is much simpler than Einstein’s theory. Moreover, it tells us that the restricted gravity can be written as an Abelian gauge theory, which implies that the graviton can be described by a massless spin-1 field. This establishes the Abelian dominance in Einstein’s theory.

Abelianization for hyperkähler quotients

An equivariantly gauge-fixed non-abelian gauge theory is a theory in which a coset of the gauge group, not containing the maximal abelian subgroup, is gauge fixed. Such theories are non-perturbatively well-defined. In a finite volume, the equivariant BRST symmetry guarantees that expectation values of gauge-invariant operators are equal to their values in the unfixed theory. However, turning on a small breaking of this symmetry, and turning it off after the thermodynamic limit has been taken, can in principle reveal new phases. In this paper we use a combination of strong-coupling and mean-field techniques to study an SU(2) Yang--Mills theory equivariantly gauge fixed to a U(1) subgroup. We find evidence for the existence of a new phase in which two of the gluons becomes massive while the third one stays massless, resembling the broken phase of an SU(2) theory with an adjoint Higgs field. The difference is that here this phase occurs in an asymptotically-free theory.

§ 252. Nonabelian p-groups all of whose maximal abelian subgroups are isolated

It has been observed by M. Takesaki (private communication) that for the C*-algebras 0 n there is a bijective correspondence between unitaries in 0 n and endomorphisms that leave the unit fixed. We use this idea to study automorphisms of 0 n. The group Aut 0 n is shown to contain a maximal abelian subgroup T (that is canonically associated with the construction of 0 n from a topological Markov.chain; cf. the paper by W. Krieger and the author: A class of C*-algebras and topological Markov chains). This maximal com- mutative subgroup of Aut 0 n is similar to a maximal torus in a semisimple Lie group in that it is an inductive limit of tori, and its Weyl group, i.e. the quotient of its normalizer in Aut 0 n by T itself is discrete.