Results 1  10
of
18
Random subgroups and analysis of the lengthbased and quotient attacks
 Journal of Mathematical Cryptology
"... ..."
(Show Context)
DECISION PROBLEMS AND PROFINITE COMPLETIONS OF GROUPS
, 2008
"... We consider pairs of finitely presented, residually finite groups P ↩ → Γ for which the induced map of profinite completions ˆ P → ˆ Γ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not P is isomorphic to Γ. We construct pairs fo ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
(Show Context)
We consider pairs of finitely presented, residually finite groups P ↩ → Γ for which the induced map of profinite completions ˆ P → ˆ Γ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not P is isomorphic to Γ. We construct pairs for which the conjugacy problem in Γ can be solved in quadratic time but the conjugacy problem in P is unsolvable. Let J be the class of superperfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group Γ and a guarantee that Γ ∈ J, can determine whether or not Γ ∼ = {1}. We construct a finitely presented acyclic group H and an integer k such that there is no algorithm that can determine which kgenerator subgroups of H are perfect.
Splittings and automorphisms of relatively hyperbolic groups
, 2012
"... We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgrou ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GLn(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GLn(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a hyperbolic group G, we view G as hyperbolic relative to P and we apply the previous analysis to describe the group Out(P 1G) of automorphisms of P that extend to G: it is virtually a McCool group. If Out(P 1G) is infinite, then P is a vertex group in a splitting of G. If P is torsionfree, then Out(P 1G) is of type VF, in particular finitely presented. We also determine when Out(G) is infinite, for G relatively hyperbolic. The interesting case is when G is infinitelyended and has torsion. When G is hyperbolic, we show that Out(G) is infinite if and only if G splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of Out(G) comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.
ON THE FINITE PRESENTATION OF SUBDIRECT PRODUCTS AND THE NATURE OF RESIDUALLY FREE Groups
"... We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of subdirect products of groups: if Γ1,..., Γn are finitely presented and S < Γ1 × · · ·×Γn projects to a subgroup of finite index in each Γi × Γj, then S is finitely presentable, indeed there i ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
(Show Context)
We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of subdirect products of groups: if Γ1,..., Γn are finitely presented and S < Γ1 × · · ·×Γn projects to a subgroup of finite index in each Γi × Γj, then S is finitely presentable, indeed there is an algorithm that will construct a finite presentation for S. We use the VSP criterion to characterise the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither FP ∞ nor of StallingsBieri type.
Polycyclic groups: a new platform for cryptography, Preprint arXiv: math.GR/0411077
, 2004
"... Abstract. We propose a new cryptosystem based on polycyclic groups. The cryptosystem is based on the fact that the word problem can be solved effectively in polycyclic groups, while the known solutions to the conjugacy problem are far less efficient. 1. ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We propose a new cryptosystem based on polycyclic groups. The cryptosystem is based on the fact that the word problem can be solved effectively in polycyclic groups, while the known solutions to the conjugacy problem are far less efficient. 1.
Polynomialtime word problems
, 2006
"... Abstract. We find polynomialtime solutions to the word problem for freebycyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We find polynomialtime solutions to the word problem for freebycyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble straight line programs, which are widely studied in the theory of compressed data structures. In an effort to be selfcontained we give a detailed exposition of the necessary results from computer science. 1.
On endomorphisms of torsionfree hyperbolic groups
, 2009
"... Let H be a torsionfree δhyperbolic group with respect to a finite generating set S. Let a1,..., an and a1∗,...,an ∗ be elements of H such that ai ∗ is conjugate to ai for each i = 1,...,n. Then, there is a uniform conjugator if and only if W(a1∗,...,an∗) is conjugate to W(a1,..., an) for every wor ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Let H be a torsionfree δhyperbolic group with respect to a finite generating set S. Let a1,..., an and a1∗,...,an ∗ be elements of H such that ai ∗ is conjugate to ai for each i = 1,...,n. Then, there is a uniform conjugator if and only if W(a1∗,...,an∗) is conjugate to W(a1,..., an) for every word W in n variables and length up to a computable constant depending only on δ, ♯S and ∑n i=1 ai. As a corollary, we deduce that there exists a computable constant C = C(δ, ♯S) such that, for any endomorphism ϕ of H, if ϕ(h) is conjugate to h for every element h ∈ H of length up to C, then ϕ is an inner automorphism. Another corollary is the following: if H is a torsionfree conjugacy separable hyperbolic group, then Out(H) is residually finite. When particularizing the main result to the case of free groups, we obtain a solution for a mixed version of the classical Whitehead’s algorithm. 1
Lectures on Geometric Group Theory
"... The goal of this book is to present several central topics in geometric group theory, primarily related to the large scale geometry of infinite groups and spaces on which such groups act, and to illustrate them with fundamental theorems such as Gromov’s Theorem on groups of polynomial growth, Tits’ ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
The goal of this book is to present several central topics in geometric group theory, primarily related to the large scale geometry of infinite groups and spaces on which such groups act, and to illustrate them with fundamental theorems such as Gromov’s Theorem on groups of polynomial growth, Tits’ Alternative, Mostow Rigidity Theorem, Stallings’ theorem on ends of groups, theorems of Tukia and Schwartz on quasiisometric rigidity for lattices in realhyperbolic spaces, etc. We give essentially selfcontained proofs of all the above mentioned results, and we use the opportunity to describe several powerful tools/toolkits of geometric group theory, such as coarse topology, ultralimits and quasiconformal mappings. We also discuss three classes of groups central in geometric group theory: Amenable groups, (relatively) hyperbolic groups, and groups with Property (T). The key idea in geometric group theory is to study groups by endowing them with a metric and treating them as geometric objects. This can be done for groups that are finitely generated, i.e. that can be reconstructed from a finite subset, via multiplication and inversion. Many groups naturally appearing in topology,
FINITELY PRESENTED RESIDUALLY FREE GROUPS
, 2008
"... We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all n ∈ N, a residually free group is of type FPn if and only if it is of type Fn. New families of subdirect pro ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all n ∈ N, a residually free group is of type FPn if and only if it is of type Fn. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither FP ∞ nor of StallingsBieri type. The template for these examples leads to a more constructive characterization of finitely presented residually free groups up to commensurability. We show that the class of finitely presented residually free groups is recursively enumerable and present a reduction of the isomorphism problem. A new algorithm is described which, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. The (multiple) conjugacy and membership problems for finitely presented subgroups of residually free groups are solved.