52 research outputs found
On the isolated points in the space of groups
We investigate the isolated points in the space of finitely generated groups.
We give a workable characterization of isolated groups and study their
hereditary properties. Various examples of groups are shown to yield isolated
groups. We also discuss a connection between isolated groups and solvability of
the word problem.Comment: 30 pages, no figure. v2: minor changes, published version from March
200
Independence in computable algebra
We give a sufficient condition for an algebraic structure to have a
computable presentation with a computable basis and a computable presentation
with no computable basis. We apply the condition to differentially closed, real
closed, and difference closed fields with the relevant notions of independence.
To cover these classes of structures we introduce a new technique of safe
extensions that was not necessary for the previously known results of this
kind. We will then apply our techniques to derive new corollaries on the number
of computable presentations of these structures. The condition also implies
classical and new results on vector spaces, algebraically closed fields,
torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page
The centre of a finitely generated strongly verbally closed group is almost always pure
The assertion in the title implies that many interesting groups (e.g., all
non-abelian braid groups or ) are not strongly
verbally closed, i.e., they embed into some finitely generated groups as
verbally closed subgroups, which are not retracts.Comment: 5 pages. A Russian version of this paper is at
http://halgebra.math.msu.su/staff/klyachko/papers.ht
Model-theoretic properties of nilpotent groups and Lie algebras
We give a systematic study of the model theory of generic nilpotent groups
and Lie algebras. We show that the Fra\"iss\'e limit of 2-nilpotent groups of
exponent studied by Baudisch is 2-dependent and NSOP. We prove that
the class of -nilpotent Lie algebras over an arbitrary field, in a language
with predicates for a Lazard series, is closed under free amalgamation. We show
that for , the generic -nilpotent Lie algebra over
is strictly NSOP and -dependent. Via the Lazard correspondence, we
obtain the same result for -nilpotent groups of exponent , for an odd
prime
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