Some Examples of Linear Groups

In 1940 Nisnevic̆ published the following theorem [3]. Let (Gα)α∈Λ be a family of groups indexed by some set Λ and (Fα)α∈Λ a family of fields of the same characteristic p⩾0. If for each α the group Gα has a faithful representation of degree n over Fα then the free product*α∈Λ Gα has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (Gα)α∈Λ ⩽GL(n, F) is a family of subgroups for which there exists Z⩽GL(n, F) such that for all α the intersection Gα∩F.1n=Z, then the free product of the groups *ZGα with Z amalgamated via the identity map is isomorphic to a linear group of degree n over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize these results from the linear to the skew-linear case, that is, to groups isomorphic to subgroups of GL(n, Dα) where the Dα are division rings. In fact, many of the results can be generalized to rings which, although not necessarily commutative, contain no zero-divisors. We have the following.

Products of Free Groups in the Unit Group of Integral Group Rings II

Free products of completely positive maps and spectral sets

Left-orderings on free products of groups

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group FA has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that FA is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and FA is a free A-groupp of Lyndon' type, then FA has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then FA has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, FA has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if FQ has a faithful Magnus representation in Q[[Y]].

Long colimits of topological groups II: Free groups and vector spaces

Let $(G_\ell\mid\ell\in L)$ be a family of groups and let $F$ be a free group. Let $G$ denote $F \ast {\Large{*}}_{\ell\in L} G_\ell$, the free product of $F$ and all the $G_\ell$. Let $\mathcal{F}$ denote the set of all finitely generated (free) subgroups $H$ of $G$ which have the property that, for each $g \in G$ and each $\ell\in L$, $H \cap G_\ell^{g} = \{1\}$. For each free group $H$, the reduced rank of $H$ is defined as $\bar{r}(H) := \max\{\operatorname{rank}(H) -1, 0\} \in[0,\infty]$. Set \begin{eqnarray*} \theta & := & \max\biggl\{ \biggl\vert\frac{\lvert D \rvert}{\lvert D \rvert-2}\biggr\vert : &&{}\mbox{$D$ is a finite subgroup of $G$ with }\lvert D \rvert \ne2 \biggr\} \in [1,3], \sigma & := & \inf\biggl\{ s \in[0,\infty] : \mbox{for all } H, K \in\mathcal{F}, &&{}\sum_{HgK \in H \setminus G /K}\bar{r} (H^g\cap K) \le s {\theta} \bar{r} (H) \bar{r} (K) \biggr\} \in [0,\infty]. \end{eqnarray*} We are interested in precise bounds for $\sigma$. If every element of $\mathcal{F}$ is cyclic, then $\sigma = 0$. We henceforth assume that some element of $\mathcal{F}$ has rank two. In the case where $G=F$ and, hence, $v = 1$, Hanna Neumann and Walter Neumann proved that $\sigma\in[1,2]$ and it is a famous conjecture that $\sigma = 1$, called the Strengthened Hanna Neumann Conjecture. For the general case, we proved that $\sigma \in[1,2]$ and if $G$ has 2-torsion then $\sigma = 2$. We conjectured that if $G$ is 2-torsion-free then $\sigma = 1$. In this article, we prove the following implications which show that under certain circumstances $\sigma \lt 2$. If $G$ is $2$-torsion-free and has $3$-torsion, then $\sigma \le \frac{8}{7}$. If $G$ is $2$-torsion-free and $3$-torsion-free and has $5$-torsion, then $\sigma \le \frac{9}{5}$. If $p$ is an odd prime number and $G = C_p \ast C_p$, then $\sigma \le 2 - \frac{(4+2\sqrt{3})p}{(2 p -3 + \sqrt{3})^2}$. In particular, if $G = C_3 \ast C_3$ then $\sigma = 1$, and if $G = C_5 \ast C_5$ then $\sigma \le 1.52$.

The purpose of this paper is to study the following two questions.(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?(2) When is the group algebra of a torsion free soluble group primitive?In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

Summary The paper presents a simple proof by induction that every word in the free product of groups has a unique reduced form. The only prerequisite is the knowledge of the definition of group.

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

We will construct an almost free non-free group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\aleph _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper G prime"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">G/G’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a free abelian group.

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

Let V be a vector space over a field F. If G ≤ GL(V, F), the central dimension of G is the F-dimension of the vector space V/C V (G). In Dixon et al. (2004) and Kurdachenko and Subbotin (2006), soluble linear groups in which the set ℒ icd(G) of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. In this article we study periodic locally radical linear groups, in which the set ℒ icd(G) satisfies one of the weak chain conditions: the weak minimal condition or the weak maximal condition.