Dennis Spellman

The Persistence of Universal Formulae in Free Algebras

Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let F α ( V ) be a V -free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements: (1) F r (V) discriminates V . (1*) The F s (V) satisfy the same universal sentences for all s ≥ r . Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent: (2) F r (V) strongly discriminates V . (2*) The F s (V) satisfy the same universal formulas for all s ≥ r whenever elements of F r (V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

Almost Locally Free Groups and a Theorem of Magnus: Some Questions

Ben Fine observed that a theorem of Magnus on normal closures of elements in free groups is first order expressible and thus holds in every elementarily free group. This classical theorem, vintage 1931, asserts that if two elements in a free group have the same normal closure, then either they are conjugate or one is conjugate to the inverse of the other in the free group. An examination of a set of sentences capturing this theorem reveals that the sentences are universal-existential. Consequently the theorem holds in the almost locally free groups of Gaglione and Spellman. We give a sufficient condition for the theorem to hold in every fully residually free group as well as a sufficient condition for the theorem to hold, even more generally, in every residually free group.

Reflections on Discriminating Groups

Abstract Here we continue the study of discriminating groups as introduced by Baumslag, Myasnikov and Remeslennikov in [G. Baumslag, A. G. Myasnikov and V. N. Remeslennikov. Discriminating and codiscriminating groups. J. Group Theory 3 (2000), 467–479.]. First we give examples of finitely generated groups which are discriminating but not trivially discriminating, in the sense that they do not embed their direct squares, and then we show how to generalize these examples. In the opposite direction we show that if F is a non-abelian free group and R is a normal subgroup of F such that F/R is torsion-free, then F/R′ is non-discriminating.

Notions of Discrimination

As an outgrowth of the study of algebraic geometry over groups, discriminating groups were introduced. Many important universal type groups such as Higmans universal group and Thompsons group F were shown to be discriminating. Squarelike groups were then introduced to better capture axiomatic properties of discrimination. In the present article squarelike groups are reinterpreted in terms of discrimination of quasi varieties, and the relationship with an older version of discrimination, termed varietal discrimination here, is studied.

The Elementary Theory of Groups: A Guide Through the Proofs of the Tarski Conjectures. Vol. 60

After being an open question for sixty years the Tarski conjecture was answered in the affirmative by Olga Kharlampovich and Alexei Myasnikov and independently by Zlil Sela. This book is an examination of the material on the general elementary theory of groups that is necessary to begin to understand the proofs.

On Some Finiteness Properties in Infinite Groups

We consider some questions concerning finiteness properties in infinite groups which are related to Marshall Halls theorem. We call these properties Property S and Property R, and they are trivially true in finite groups. We give several elementary proofs using these properties for results on finitely generated subgroups of free groups as well as a new elementary proof of Halls basic result. Finally, we consider these properties within surface groups and prove an analog of Halls theorem in that context.

Orderable groups, elementary theory, and the Kaplansky conjecture

Abstract We show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that 𝒦 {{\mathcal{K}}} , the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in 𝒦 {{\mathcal{K}}} or more generally two torsion-free groups are universally equivalent.

On CT and CSA groups and related ideas

Abstract A group is G commutative transitive or CT if commuting is transitive on nontrivial elements. A group G is CSA or conjugately separated abelian if maximal abelian subgroups are malnormal. These concepts have played a prominent role in the studies of fully residually free groups, limit groups and discriminating groups. They also play a role in the solution to the Tarski problems. CSA always implies CT however the class of CSA groups is a proper subclass of the class of CT groups. For limit groups and finitely generated elementary free groups they are equivalent. In this paper we examine the relationship between the two concepts. In particular, we show that a finite CSA group must be abelian. If G is CT, then we prove that G is not CSA if and only if G contains a nonabelian subgroup G 0

Discrimination in a General Algebraic Setting

{G_{0}}

An application of elementary real analysis to a metabelian group admitting integral polynomial exponents

which contains a nontrivial abelian subgroup H that is normal in G 0