56 research outputs found
On Residual Properties of Word Hyperbolic Groups
For a fixed word hyperbolic group we compare different residual properties
related to quasiconvex subgroups.Comment: 17 pages. Final version: several typos correcte
New examples of groups acting on real trees
We construct the first example of a finitely generated group which has
Serre's property (FA) (i.e., whenever it acts on a simplicial tree it fixes a
vertex), but admits a fixed point-free action on an -tree with
finite arc stabilizers. We also give a short and elementary construction of
finitely generated groups that have property (FA) but do not have
(F).Comment: v4: 22 pages, revised following referee's suggestions. This version
of the paper has been accepted by the Journal of Topolog
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
On subgroups of right angled Artin groups with few generators
For each natural number we construct a -generated group , which
is a subdirect product of free groups, such that the cohomological dimension of
is . Given a group and a normal subgroup we prove that
any right angled Artin group containing the special HNN-extension of with
respect to must also contain . We apply this to construct, for every
, a -generated group , embeddable into a right angled
Artin group, such that the cohomological dimension of is but the
cohomological dimension of any right angled Artin group, containing , is
at least . These examples are used to show the non-existence of certain
"universal" right angled Artin groups.
We also investigate finitely presented subgroups of direct products of limit
groups. In particular we show that for every there exists
such that any -generated finitely presented
subgroup of a direct product of finitely many free groups embeds into the
-th direct power of the free group of rank . As another corollary
we derive that any -generated finitely presented residually free group
embeds into the direct product of at most limit groups.Comment: v4: accepted in this format for publication in Intern. J. Algebra and
Comput.; 12 page
Tits alternatives for graph products
We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products
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
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