167 research outputs found
Homogeneity and prime models in torsion-free hyperbolic groups
We show that any nonabelian free group of finite rank is homogeneous;
that is for any tuples , , having the same complete
-type, there exists an automorphism of which sends to .
We further study existential types and we show that for any tuples , if and have the same existential -type,
then either has the same existential type as a power of a primitive
element, or there exists an existentially closed subgroup (resp.
) of containing (resp. ) and an isomorphism
with .
We will deal with non-free two-generated torsion-free hyperbolic groups and
we show that they are -homogeneous and prime. This gives, in
particular, concrete examples of finitely generated groups which are prime and
not QFA
Ampleness in the free group
We show that the theory of the free group -- and more generally the theory of
any torsion-free hyperbolic group -- is -ample for any . We give
also an explicit description of the imaginary algebraic closure in free groups
The monomorphism problem in free groups
Let F be a free group of finite rank. We say that the monomorphism problem in F is decidable if there is an algorithm such that, for any two elements u and v in F, it determines whether there exists a monomorphism of F that sends u to v. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the proble
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