53 research outputs found

    Recognizing a relatively hyperbolic group by its Dehn fillings

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    Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic 33-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.Comment: Minor modification (including typesetting). 56 page

    Algorithmic properties of poly-Z groups and secret sharing using non-commutative groups

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    Computational aspects of polycyclic groups have been used to study cryptography since 2004 when Eick and Kahrobaei proposed polycyclic groups as a platform for conjugacy based cryptographic protocols. In the first chapter we study the conjugacy problem in polycyclic groups and construct a family of torsion-free polycyclic groups where the uniform conjugacy problem over the entire family is at least as hard as the subset sum problem. We further show that the conjugacy problem in these groups is in NP, implying that the uniform conjugacy problem is NP-complete over these groups. This is joint work with Delaram Kahrobaei. We also present an algorithm for the conjugacy problem in groups of the form Znâ‹ŠĎ•Z\Z^n \rtimes_\phi \Z. We continue by studying automorphisms of poly-Z\Z groups and successive cyclic extensions of arbitrary groups. We study a certain kind of extension that we call deranged , and show that the automorphisms of the resulting group have a strict form. We also show that the automorphism group of a group obtained by iterated extensions of this type contains a non-abelian free group if and only if the original base group does. Finally we show that it is possible to verify that a finitely presented by infinite cyclic group is finitely presented by infinite cyclic, but that determining that a general finitely presented group is finitely generated by infinite cyclic is undecidable. We then discuss implications the latter result has for calculating the Bieri-Neumann-Strebel invariant. This is joint work with Jordi Delgado, Delaram Kahrobaei, Ha Lam, and Enric Ventura and is currently in preparation. In the final chapter we discuss secret sharing schemes and variations. We begin with classical secret sharing schemes and present variations that allow them to be more practical. We then present a secret sharing scheme due to Habeeb, Kahrobaei, and Shpilrain. Finally, we present an original adjustment to their scheme that involves the shortlex order on a group and allows less information to be transmitted each time a secret is shared. Additionally, we propose additional steps that allow participants to update their information independently so that the scheme remains secure over multiple rounds. This is joint work with Delaram Kahrobaei

    On geometric aspects of diffuse groups

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    Bowditch introduced the notion of diffuse groups as a geometric variation of the unique product property. We elaborate on various examples and non-examples, keeping the geometric point of view from Bowditch's paper. In particular, we discuss fundamental groups of flat and hyperbolic manifolds. The appendix settles an open question by providing an example of a group which is diffuse but not left-orderable.Comment: 37 pages, main text by Kionke and Raimbault, appendix by Dunfield. v2 : updated ancillary files and added url to the reference

    TC^0 Circuits for Algorithmic Problems in Nilpotent Groups

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    Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC^0 - uniformly for all r-generated nilpotent groups of class at most c for fixed r and c. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform TC^0, while all the other problems we examine are shown to be TC^0-Turing reducible to the problem of computing greatest common divisors and expressing them as linear combinations

    Algebraic properties of profinite groups

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    Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update
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