Forcing a basis into ℵ1-free groups Bossaller, Daniel; Herden, Daniel; Pasi, Alexandra V.
Journal of algebra,
08/2023, Letnik:
627
Journal Article
Recenzirano
In this paper, we address the question of when a non-free ℵ1-free group H can be free in a transitive cardinality-preserving model extension. Using the Γ-invariant, denoted Γ(H), we present a ...necessary and sufficient condition resolving this question for ℵ1-free groups of cardinality ℵ1. Specifically, if Γ(H)=ℵ1, then H will be free in a transitive model extension if and only if ℵ1 collapses, while for Γ(H)≠ℵ1 there exist cardinality-preserving forcings that will add a basis to H. In particular, for Γ(H)≠ℵ1, we provide a poset (Ppb,⩽) of partial bases for adding a basis to H without collapsing ℵ1.
We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, ...for every basis of the ambient group. In the course of the proof we present a partition of the set of homomorphisms between free groups which is of independent interest.
Let G be a group. An element g∈G is called a test element of G if for every endomorphism φ:G→G, φ(g)=g implies that φ is an automorphism. Let F(X) be a free group on a finite non-empty set X, and let ...X=X1∐X2∐…∐Xr be a finite partition of X into r≥2 non-empty subsets. For i=1,2,…,r, let ui∈〈Xi〉≤F(X), and let w(z1,…,zr) be a word in the variables z1,…,zr. We give several sufficient conditions on ui (1≤i≤r) and w for w(u1,…,ur) to be a test element of F(X). As an application of these results, we give examples of test elements of a free group of rank greater than two that are not test elements in any pro-p completion of the group.
Every word in a free group F induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, ...necessarily belong to the same orbit of AutF. A special case of this problem, when one of the words is the primitive word x, was settled positively by the third author and Parzanchevski 15. Here we extend this result to the case where one of the words is xd or x,yd for an arbitrary d∈Z.
In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let F be a free group with basis X and let r be a cyclically reduced element of F which contains a basis element x∈X, then every ...non-trivial element of the normal closure of r in F contains the basis element x. Equivalently, the subgroup freely generated by X﹨{x} embeds canonically into the quotient group F/〈〈r〉〉F. In this article, we want to introduce a Freiheitssatz for amalgamated products G=A⁎UB of free groups A and B, where U is a maximal cyclic subgroup in A and B: If an element r of G is neither conjugate to an element of A nor B, then the factors A, B embed canonically into G/〈〈r〉〉G.
We show that many 2-dimensional Artin groups are residually finite. This includes 3-generator Artin groups with labels ≥4 except for (2m+1,4,4) for any m≥2. As a first step towards residual ...finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.
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