Almost locally free groups and the genus question
作者:
Anthony M. Gaglione,
Dennis Spellman,
期刊:
Communications in Algebra
(Taylor Available online 1998)
卷期:
Volume 26,
issue 9
页码: 2821-2836
ISSN:0092-7872
年代: 1998
DOI:10.1080/00927879808826312
出版商: Gordon and Breach Science Publishers Ltd.
数据来源: Taylor
摘要:
Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F2)⋂∀∃ in a first-order language Lo appropriate for group theory. It is shown that in every model of Th(F2)⋂∀∃ the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ∃-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F2)⋂∀∃ while the quasi-locally free groups are the ∃-free groups with maximal Abelian subgroups elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound of finitely generated quasi-locally free groups containing nontrivial torsion in their Abelianizations. The question of whether or not almost locally free groups have torsion free Abelianization is related to a bound in a free group on the number of factors needed to express certain elements of the derived group as a product of commutators
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