Finding bases of uncountable free abelian groups is usually difficult

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

Several authors have recently studied the problem of finding circumstances under which the amenability of a semigroup is inherited by a subsemigroup. It is known ([2] and [3]) that subgroups of an amenable group are amenable, but that in general subsemigroups of an amenable semigroup are not. We show here that subsemigroups of an amenbable group need not be amenable. Frey proved in [4] that subsemigroups of a group inherit amenability and the group contains no copy of the free semigroup S on two generators (S is not amenable). The problem is discussed in detail in [51. It is known [11 that a group generated by a copy of S contained in it need not be free; the question is whether it can be far enough from free to be amenable. Since solvable groups are amenable [6], our claim that subsemigroups of amenable groups need not be amenable will be established we can embed S in a solvable group. We proceed to do this. Granirer conjectured in his review [MR 35 #3423] of Wilde and Witz [7] that a group is amenable and it contains no copy of S. It will follow that the only if part of this conjecture is quite false. Let B be the free additive abelian group on the family of generators {b(m, n) }, m, n integers. Any permutation of these generators induces an automorphism of B. Let x and y be the automorphisms induced by b(m, n)F->b(m+1, n) and b(m, n)'->b(m, n+1), respectively. It is easy to see that the subgroup C of Aut B generated by x and y is the free abelian group on the generators x, y. We write C additively, and we indicate the action of an element c of C on an element b of B by be. Let be the normal product of B by C. consists of pairs [c, b] and the multiplication is given by -[c, b1i [c, b] [cl+c, b1+b]. c1->[c, 0], bF-*[O, b] are embeddings of C, B respectively into G, and B is thus a normal abelian subgroup with quotient G/B-C. Hence, is solvable and in fact the second derived group G of is trivial. The same holds for every subgroup of G. Now, let s= [x, b(O, 0)] and t= [y, b(0, 0)].

Given an N 1 -free abelian group G we characterize the class? G of all torsion abelian groups T satisfying Ext(G,T) = 0 assuming the special continuum hypothesis CH. Moreover, in Godel's constructible universe we prove that this characterizes? G for arbitrary torsion-free abelian G. It follows that there exists some ugly N 1 -free abelian groups.

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

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

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

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 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.

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.

On λ-homomorphic skew braces

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.

I have developed an algorithm and written a GAP functions pautfreegroup:=function(n) (Finite Presentation of automorphism Groups of Free group of Rank n), and autfreeabeliangroup:=function(n) (Finite Presentation of automorphism Groups of Free abelian group of Rank n). Functions using GAP system for computation of a finite presentation for Aut(F_n ) and GL(n,z) respectively. In order to do that we have given a description for the presentation of Aut(F_n) and GL(n,z), the automorphism groups of free groups and free abelian groups respectively.

Let T be an abelian group and λ an uncountable regular cardinal. The question of whether there is a λ-universal group U among all torsion-free abelian groups G of cardinality less than or equal to λ satisfying Exta(G,T) = 0 is considered. U is said to be λ-universal for T if, whenever a torsion-free abelian group G of cardinality at most λ satisfies Ext(G,T)=0, there is an embedding of G into U. For large classes of abelian groups T and cardinals λ, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and λ, there is no universal group. In particular, for T torsion, this solves a problem by Kulikov.

This Note investigates certain properties of torsion--free abelian groups which are related to types. We start by intro-ducing for every type t two functorial subgroups of G called G[t], G *[t], which are in some way duals to the classical G(t), G *(t). Paragraph 1 is dedicated to studying the general properties of the new subgroups, their relations to the old ones, and those properties of G which naturally follow from these relations. For instance, we are lead to the source, in G, of rank 1, type t summands of G. In para- graph 2, by slightly strenghthening two of the general properties obtained, we get a class c of t.f. groups which, besides having some nice closure properties and containing separable groups and vector groups, is defined “locally at the type t”: i.e. to see that G E c one studies the behaviour of G w.r. to type t “one type at a time”. Finally, in Paragraph 3, by introducing and investigating separability as an element property rather than a group property, and by using “localization at the type t”, we get as a bonus a very simple proof of the fundamental result by Fuchs stating that summands of separable groups are separable.

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.