19 research outputs found
Separating cyclic subgroups in graph products of groups
We prove that the property of being cyclic subgroup separable, that is having
all cyclic subgroups closed in the profinite topology, is preserved under
forming graph products.
Furthermore, we develop the tools to study the analogous question in the
pro- case. For a wide class of groups we show that the relevant cyclic
subgroups - which are called -isolated - are closed in the pro- topology
of the graph product. In particular, we show that every -isolated cyclic
subgroup of a right-angled Artin group is closed in the pro- topology, and
we fully characterise such subgroups.Comment: 37 pages, revised following referee's comments, to appear in Journal
of Algebr
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
Combinatorial group theory and cryptography
In the presented work we focus on applications of decision problems from combinatorial group theory. Namely we analyse the Shpilrain-Zapata pro- tocol. We give formal proof that small cancellation groups are good platform for the protocol because the word problem is solvable in linear time and they are generic. We also analyse the complexity of the brute force attack on the protocol and show that in a theoretical way the protocol is immune to attack by adversary with arbitrary computing power
Ireducibilní polynomy nad
Katedra algebryDepartment of AlgebraFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult
Combinatorial group theory and cryptography
In the presented work we focus on applications of decision problems from combinatorial group theory. Namely we analyse the Shpilrain-Zapata pro- tocol. We give formal proof that small cancellation groups are good platform for the protocol because the word problem is solvable in linear time and they are generic. We also analyse the complexity of the brute force attack on the protocol and show that in a theoretical way the protocol is immune to attack by adversary with arbitrary computing power