A Note on Zero Divisors in Group-Rings

We consider twisted conjugacy classes and the R ∞-property for classical linear groups. In particular, it is stated that the general linear group GL n (K) and the special linear group SL n (K), where $$ n \geqslant 3 $$ , possess the R ∞-property if either K is an infinite integral domain with trivial automorphism group or K is an integral domain containing a subring of integers, whose automorphism group Aut(K) is finite. By an integral domain we mean a commutative ring with identity which has no zero divisors.

Let be a group ring of a group G over a field K. The Ore condition says that for any there exist such that au = bv, where or It always holds whenever G is amenable. Recently it was shown that for R. Thompson’s group F the converse is also true. So the famous amenability problem for F is equivalent to the question on the Ore condition for the group ring of the same group. It is easy to see that the problem on the Ore condition for is equivalent to the same property for the monoid ring where M is the monoid of positive elements of F. In this paper we reduce the problem to the case when a, b are homogeneous elements of the same degree in the monoid ring. We study the case of degree 1 and find solutions of the Ore equation. For the case of degree 2, we study the case of linear combinations of monomials from This set is not doubling, that is, there are nonempty finite subsets such that As a consequence, the Ore condition holds for linear combinations of these monomials. We give an estimate for the degree of u, v in the above equation. The case of monomials of higher degree is open as well as the case of degree 2 for monomials on where Recall that negative answer to any of these questions will immediately imply non-amenability of F.

Three theorems concerning the form ring (= associated graded ring) F ( R , I ) {\mathbf {F}}\left ( {R,I} \right ) of an ideal I I in a Noetherian ring R R are proved. The first characterizes, for a P P -primary ideal in a locally quasi-unmixed ring, when F ( R , I ) red {\mathbf {F}}{\left ( {R,I} \right )_{\operatorname {red}}} is an integral domain in terms of when F ( R P , I R P ) red {\mathbf {F}}{\left ( {{R_P},I{R_P}} \right )_{\operatorname {red}}} is an integral domain. For an aribtrary Noetherian ring R R the second gives a somewhat similar characterization for F ( R , J ) {\mathbf {F}}\left ( {R,J} \right ) to have only one prime divisor of zero for some ideal J J that is projectively equivalent to I I . And the third characterizes unmixed semilocal rings in terms of the existence of an open ideal I I such that the zero ideal in F ( R , I ) {\mathbf {F}}\left ( {R,I} \right ) is isobathy.

A central element of the complex group algebra C G {\mathbf {C}}G which is a zero divisor in the W ∗ {W^ * } group algebra W ( G ) W(G) is also a zero divisor in C G {\mathbf {C}}G . As a corollary, if K K is a field of characteristic zero, G G is a group, A A is an abelian normal subgroup of G G , and R R is the Ore localization of K G KG obtained by inverting all nonzero elements of K A KA , then all matrix rings over R R are directly finite and R R has the invariant basis property.

Dedication Preface Historical perspective Conference participants Conference photo Conference timetable 1. On flatness and the Ore condition J. A. Beachy 2. Localization in general rings, a historical survey P. M. Cohn 3. Noncommutative localization in homotopy theory W. G. Dwyer 4. Noncommutative localization in group rings P. A. Linnell 5. A non-commutative generalisation of Thomason's localisation theorem A. Neeman 6. Noncommutative localization in topology A. A. Ranicki 7. L2-Betti numbers, isomorphism conjectures and noncommutative localization H. Reich 8. Invariants of boundary link cobordism II. The Blanchfield-Duval form D. Sheiham 9. Noncommutative localization in noncommutative geometry Z. Skoda.

A group G which can be defined by a single relation in which there are no negative exponents is residually solvable. If G is also torsion-free then it is locally indicable and hence its integral group ring has no zero divisors.

We study the zero divisor problem for group rings of some classes of wreath products. We prove also that some of these group rings embed into division rings or prime Artinian rings.

By a group ring we mean in this paper a ring defined by a finite group G and an integral domain K:such that A contains G and is freely generated by G over K, so thatThe ring A = KG has a co-multiplicationdefined byso that A is a Hopf algebra.

The computation of the knot cobordism groups reduces to the computation of the L-theory of various fields and integral domains. The main properties of the L-groups of a field with involution F are summarized in 37A. The computations of the L-groups of ℤ, ℤ m , \({\widehat {\Bbb Z}_m}\), ℚ and the localization exact sequence are used in 37B to prove a general result: the natural maps L* (A) → L* (ℚ ⊗ℤ A) are isomorphisms modulo 8-torsion for any additively torsion-free ring involution A (e.g. a group ring A = ℤ[π]). In Chap. 42 the version of this result for torsion L-groups will be used to prove that the natural maps $${C_*} = LIs{o^{* + 1}}({\Bbb Z}) \to LIso_{{U_{\Bbb Z}}}^{* + 1}({\Bbb Q})$$ are isomorphisms modulo 8-torsion, for an appropriately defined U ℤ.

It is proved that from the isomorphism of the integral group rings of two groups follows the isomorphism of the quotient-groups of these groups by the second commutator subgroup. Groups with certain conditions T1 and T2 are considered. From condition T1 it follows that the group algebra over an integral domain has no zero-divisors, and from T2 that the group of invertible elements of this algebra is trivial. Sufficient conditions are given, under which there are no zero-divisors in the group algebra of a pro-p-group over a field of p elements.

The importance of the role played by free products in combinatorial group theory is no less than that of direct products. The topological motivation is that the free product F of a set of groups G1,G2,… is the fundamental group of the bouquet of spaces with fundamental groups G1,G2,… [154]. If an arbitrary group G is generated by subgroups G1,G2,…, then it is canonically represented as a quotient group of the free product F. Thus, G can be presented using relations needed to define this quotient group. Lyndon [146], [149] formulated an analogue of van Kampen’s lemma for free products and applied it to small cancellation free products. In Chapter 11, we extend the method and the techniques of Chapters 4–10 to diagrams over free products and apply them to quotient groups of free products.KeywordsSimple GroupMaximal SubgroupFree ProductQuotient GroupProper SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A group is said to be locally indicable if each of its non-trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image. Such groups were studied by Higman [-9] in connection with the zero-divisor and unit problems for group rings. More recently, they have arisen I-2, 12] in the study of equations over groups. In [12], I proved a Freiheitsatz for locally indicable groups. This has been proved independently by Brodskii [2] and Short [22]. The present paper arises from an investigation of a question put to me by S.J. Pride whether torsionfree 1-relator groups are locally indicable. The question was raised originally by Baumslag (I-1], Problem 19) and an affirmative solution has recently been announced by Brodski~ [2]. As a consequence, the group algebra RG of a torsion-free 1-relator group G over an integral domain R has no non-trivial zero-divisors, and no non-trivial units (using Higman's results [9]). The first fact was also proved by Lewin and Lewin [16], by embedding RG in a division ring. The second appears to be new, as was pointed out to me by K.A. Brown. A second consequence is that no 1-relator group has a non-trivial finitely generated perfect subgroup, which answers [1], Problem 7 and [10], Question 1. In fact, using the Freiheitsatz, and the tower method described in [12], it is possible to prove the following general version of Brodskii's theorem.

Let $R$ be an integral domain and $G$ be a subgroup of its group of units. We consider the category $\mathbf{\mathsf{Cob}}_G$ of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in $G$. Under some mild conditions on $R$, we construct a monoidal from $\mathbf{\mathsf{Cob}}_G$ to the category $\mathbf{\mathsf{pLagr}}_R$ consisting of pointed Lagrangian relations between skew-Hermitian $R$-modules. We call it the functor since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a $G$-equivariant version of a TQFT-like that has been described by Donaldson. The study and computation of the Magnus is carried out using classical techniques of low-dimensional topology. When $G$ is a free abelian group and $R = Z[G]$ is the group ring of $G$, we relate the Magnus to the Alexander functor (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter.

A group is called matricial field (MF) if it admits finite-dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of MF groups by showing that given two amenable groups with a common normal subgroup, the amalgamated free product is MF.

Bogomolov multipliers and retract rationality for semidirect products