369 research outputs found
-free groups are CAT(0)
We show that every group with free -length function is CAT(0).Comment: To be published in the Journal of the London Mathematical Society.
This version is very close to the accepted version. The exposition greatly
improved due to the referee's comment
On conjugacy separability of fibre products
In this paper we study conjugacy separability of subdirect products of two
free (or hyperbolic) groups. We establish necessary and sufficient criteria and
apply them to fibre products to produce a finitely presented group in
which all finite index subgroups are conjugacy separable, but which has an
index overgroup that is not conjugacy separable. Conversely, we construct a
finitely presented group which has a non-conjugacy separable subgroup of
index such that every finite index normal overgroup of is conjugacy
separable. The normality of the overgroup is essential in the last example, as
such a group will always posses an index overgroup that is not
conjugacy separable.
Finally, we characterize -conjugacy separable subdirect products of two
free groups, where is a prime. We show that fibre products provide a
natural correspondence between residually finite -groups and -conjugacy
separable subdirect products of two non-abelian free groups. As a consequence,
we deduce that the open question about the existence of an infinite finitely
presented residually finite -group is equivalent to the question about the
existence of a finitely generated -conjugacy separable full subdirect
product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio
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