3,299 research outputs found
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
Conjugacy in normal subgroups of hyperbolic groups
Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G.
We establish criteria for N to have solvable conjugacy problem and be conjugacy
separable in terms of the corresponding properties of G/N. We show that the
hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction
is hereditarily conjugacy separable. We then use this construction to produce
first examples of finitely generated and finitely presented conjugacy separable
groups that contain non-(conjugacy separable) subgroups of finite index.Comment: Version 3: 18 pages; corrected a problem with justification of
Corollary 8.
One-relator groups with torsion are conjugacy separable
We prove that one-relator groups with torsion are hereditarily conjugacy
separable. Our argument is based on a combination of recent results of Dani
Wise and the first author. As a corollary we obtain that any quasiconvex
subgroup of a one-relator group with torsion is also conjugacy separable.Comment: 9 page
Hereditary conjugacy separability of right angled Artin groups and its applications
We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy proble
Bianchi groups are conjugacy separable
We prove that non-uniform arithmetic lattices of and in
particular the Bianchi groups are conjugacy separable. The proof based on
recent deep results of Agol, Long, Reid and Minasyan.Comment: to appear in J. Pure and Applied Algebr
Effective Twisted Conjugacy Separability of Nilpotent Groups
This paper initiates the study of effective twisted conjugacy separability
for finitely generated groups, which measures the complexity of separating
distinct twisted conjugacy classes via finite quotients. The focus is on
nilpotent groups, and our main result shows that there is a polynomial upper
bound for twisted conjugacy separability. That allows us to study regular
conjugacy separability in the case of virtually nilpotent groups, where we
compute a polynomial upper bound as well. As another application, we improve
the work of the second author by giving a precise calculation of conjugacy
separability for finitely generated nilpotent groups of nilpotency class 2.Comment: V2: removed reference to false result of other paper. Accepted for
publication in Math.
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