Some primitive group rings

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.

S. K. Sehgal ([9], Problem 26) proposed the following question : Let A, B be rings and X an infinite cyclic group. Does AX ⋍ BX imply A ⋍ B? M. M. Parmenter and S. K. Sehgal (cf. [9], Chapter 4) proved that, under some strong assumptions concerning rings A, B, the answer is affirmative. In this paper, we show that the assumptions concerning the ring B may be omitted in the above mentioned results. Moreover, it is proven that if (AX)X ⋍ BX then AX ⋍ B for all rings A, B. If A is commutative and noetherian then a partial answer to Problem 27, [9] follows from our results.Recently, L. Griinenfelder and M. M. Parmenter constructed nonisomorphic rings A, B for which the group rings AX, BX are isomorphic, [2], We give a new class of rings of this type. Our examples are noncommutative domains and algebras over finite fields. That also gives a negative answer to Problem 29, [9].

We show that PD4-complexes with free fundamental group are determined by their intersection pairings and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. The purpose of this article is to show that some of the basic properties of PD4-complexes with free fundamental group can be derived homologically, without reference to the topology of 4-manifolds or stabilization by connected sums, as used in (4, 8, 13). We also avoid explicit calculations of obstructions, relying instead on the easily verified fact that the 3-skeletons of the complexes considered have su‰ciently many self homotopy equivalences. In particular we give a new proof of the fact that such complexes are determined by their intersection pairings, and that every hermitian form on a finitely generated free module over the group ring of a free group is realized by some such complex. In the final section we consider briefly the classification (up to s-cobordism) of closed 4-manifolds with free fundamental group.

Primitive group rings of soluble groups

We show that if a countable group G G is the free product of infinite abelian groups, then for every free, probability-measure-preserving (p.m.p.) action of G G , its orbit equivalence class is weakly dense in the space of p.m.p. actions of G G . This extends Lewis Bowen’s result for free groups.

Generalized Burnside–Grothendieck ring functor and aperiodic ring functor associated with profinite groups

A dichotomy for integral group rings via higher modular groups as amalgamated products

The Whitehead asphericity conjecture claims that if ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is an aspherical group presentation, then for every S ⊂ R \mathcal {S} \subset \mathcal {R} the subpresentation ⟨ A ‖ S ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {S} \, \rangle is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction of almost aspherical presentations. It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture and holds for standard presentations of free Burnside groups of large odd exponent, Tarski monsters and some others. Next, it is proven that if the Whitehead conjecture is false, then there is an aspherical presentation E = ⟨ A ‖ R ∪ z ⟩ E = \langle \, \mathcal {A} \, \| \, \mathcal {R} \cup z \, \rangle of the trivial group E E , where the alphabet A \mathcal {A} is finite or countably infinite and z ∈ A z \in \mathcal {A} , such that its subpresentation ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite A \mathcal {A} and R \mathcal {R} ), then there is a finite aspherical presentation ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle , R = { R 1 , R 2 , … , R n } \mathcal {R} = \{ R_{1}, R_{2}, \dots , R_{n} \} , such that for every S ⊆ R \mathcal {S} \subseteq \mathcal {R} the subpresentation ⟨ A ‖ S ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {S} \, \rangle is aspherical and the subpresentation ⟨ A ‖ R 1 R 2 , R 3 , … , R n ⟩ \langle \, \mathcal {A} \, \| \, R_{1}R_{2}, R_{3}, \dots , R_{n}\, \rangle of aspherical ⟨ A ‖ R 1 R 2 , R 2 , R 3 , … , R n ⟩ \langle \, \mathcal {A} \, \| \, R_{1}R_{2}, R_{2}, R_{3}, \dots , R_{n}\, \rangle is not aspherical. Now suppose a group presentation H = ⟨ A ‖ R ⟩ H = \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is aspherical, x ∉ A x \not \in \mathcal {A} , W ( A ∪ x ) W(\mathcal {A} \cup x) is a word in the alphabet ( A ∪ x ) ± 1 (\mathcal {A} \cup x)^{\pm 1} with nonzero sum of exponents on x x , and the group H H naturally embeds in G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle . It is conjectured that the presentation G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle is aspherical if and only if G G is torsion free. It is proven that if this conjecture is false and G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle is a counterexample, then the integral group ring Z ( G ) \mathbb {Z}(G) of the torsion free group G G will contain zero divisors. Some special cases where this conjecture holds are also indicated.

We prove that a countable group G can be embedded into a two-generator simple group S which is an amalgamated free product of groups G * F and F, where F and F are free groups on two generators.S is also the product of two commuting free subgroups.If G has solvable word problem, then we can construct a recursive presentation for S.1. Introduction.It is well known that every countable group is a subgroup of a simple two-generator group (see [3] and [7]).The main result of our paper is Theorem A. (a) A countable group G can be embedded into a simple two-generator group S which is an amalgamated free product of G * Fx and F, where F and Fx are both free groups on two generators.S is also the product of two commuting free subgroups.(b) If G has solvable word problem, we can construct the group S of (a) with a recursive presentation.It was first shown by R. J. Thompson [8] by a construction completely different from ours that a countable group with solvable word problem is a subgroup of a recursively presented finitely generated simple group.Our construction of the group S is similar to the construction of the simple free products of R. Camm [1], or see [6] for a slightly simplified version.2. The construction of the group S. Let 77 be a countable infinite group with an element x of order at least 5, and assume that the elements of 77 are numbered by the integers, 77 = {g" i G Z}, such that g0 = 1, gx = x, g2 = x2, g3 = x3, g4 = x4.Let A = (a) and F = (b,c) be free groups on a and b, c, respectively.Then the subgroup U of A * H generated by the elements of the form ", = aW *'eZ-{0}, where p is a permutation of Z -{0}, is a free group freely generated by the elements u" / g Z -{0}, with transversal AinA*H.Similarly, the elements Vj = bjcu\ 7GZ-{0), of F, where a is a second permutation of Z -{0}, freely generate a free subgroup V of F with transversal B = (b).

Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ℤ(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.

Let G be a complete monomial group with abelian base, namely, G = AwrSym m , the wreath product of a finite abelian group A with the symmetric group on m letters. Then the group G is determined by its integral group ring.

In this brief note, we study algebraic elements in the complex group algebra C [ G ] {\mathbf {C}}[G] . Specifically, suppose ξ ∈ C [ G ] \xi \in {\mathbf {C}}[G] satisfies f ( ξ ) = 0 f(\xi ) = 0 for some nonzero polynomial f ( x ) ∈ C [ x ] f(x) \in {\mathbf {C}}[x] . Then we show that a certain fairly natural function of the coefficients of ξ \xi is bounded in terms of the complex roots of f ( x ) f(x) . For G G finite, this is a recent observation of [HLP]. Thus the main thrust here concerns infinite groups, where the inequality generalizes results of [K] and [W] on traces of idempotents.

Let G be a simple group of Lie type over an infinite locally finite field F. For any field K, we prove that the group algebra K [ G ] K[G] is semiprimitive. The argument here is a mixture of combinatorial and topological methods. Combined with earlier results, it now follows that any group algebra of an infinite locally finite simple group is semiprimitive. Furthermore, if the group is countably infinite, then the group algebra is primitive. In particular, if G is a simple group of Lie type over the field F, then K [ G ] K[G] is a primitive ring.