Cyclic conjugacy separability and conjugacy separability of certain HNN extensions

Abstract

In this note, we give a criterion for certain HNN extensions of cyclic conjugacy separable (respectively conjugacy separable) groups with infinite cyclic associated subgroups to be again cyclic conjugacy separable (respectively conjugacy separable).

Communicated by Alexander Olshanskii     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

Conjugacy Length in Group Extensions

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Conjugacy of Morse Functions on Surfaces with Values on a Straight Line and Circle

We investigate the conjugacy of Morse functions on closed surfaces. By using cellular decompositions of surfaces, we formulate a criterion for the conjugacy of Morse functions. We establish a criterion for the conjugacy of mappings into a circle with nondegenerate critical points.     

Using Mathematica to compute conjugacy classes

The conjugacy classes of finite groups play an important role in the representation theory of those groups, and it is useful to be able to compute the conjugacy classes quickly. A procedure is developed and then implemented with Mathematica to discover these conjugacy classes. The computations make use of the Cayley table in its regular form for the group. The conjugacy classes for C_4v, the point symmetry group of the square, are displayed.     

A New Characterization of the Mathieu Groups and Alternating Groups with Degrees Primes

In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.     

Rigidity of measurable structure for \mathbb Z^d-actions by automorphisms of a torus

We show that for certain classes of actions of , by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we construct various examples of -actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure-theoretic invariant. Received: March 12, 2002     

Towards generating secure keys for braid cryptography

Braid cryptosystem was proposed in CRYPTO 2000 as an alternate public-key cryptosystem. The security of this system is based upon the conjugacy problem in braid groups. Since then, there have been several attempts to break the braid cryptosystem by solving the conjugacy problem in braid groups. In this article, we first survey all the major attacks on the braid cryptosystem and conclude that the attacks were successful because the current ways of random key generation almost always result in weaker instances of the conjugacy problem. We then propose several alternate ways of generating hard instances of the conjugacy problem for use braid cryptography.      

Parity features for classes of the infinite symmetric group

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Twisted conjugacy classes of the unit element

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Sign conjugacy classes of the alternating groups

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or ?1 on C. In this paper, we classify the sign conjugacy classes of alternating groups.     

On Reflection Subgroups of Finite Coxeter Groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.     

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The conjugacy problem for automorphism groups of countable homogeneous structures

We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.     

The Reidemeister Number of Any Automorphism of a Gromov Hyperbolic Group is Infinite

It is shown that the number of twisted conjugacy classes is infinite for any automorphism of a nonelementary, Gromov hyperbolic group. An analog of the Selberg theory for twisted conjugacy classes is suggested. Bibliography: 15 titles.     

On conjugacy in groups of finite-state automorphisms of rooted trees

We show that the conjugacy of elements of finite order in the group of finite-state automorphisms of a rooted tree is equivalent to their conjugacy in the group of all automorphisms of the rooted tree. We establish a criterion for conjugacy between a finite-state automorphism and the adding machine in the group of finite-state automorphisms of a rooted tree of valency 2. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1357–1366, October, 2008.     

A solvable conjugacy problem for finitely presented semigroups satisfying C(2) and T(4)

There are multiple, inequivalent, definitions for conjugacy in semigroups. In Cummings and Jackson (Semigroup Forum 88, 52–66, 2014), we conjectured that, for at least one of these definitions of conjugacy, the conjugacy problem for finitely presented semigroups satisfying C(2) and T(4) is solvable. Here we essentially verify that conjecture. In that 2014 Semigroup Forum publication, we developed geometric methods to solve a conjugacy problem for finitely presented semigroups satisfying C(3). We use those methods again here.     

Conjugacy separability of Seifert 3-manifold groups over non-orientable surfaces

Scott (1978) [12] showed Seifert 3-manifold groups are subgroup separable. Niblo (1992) [9] improved this result by showing that these groups are double coset separable. In Allenby, Kim and Tang (2005) [2] it was shown that all but two types of groups in the orientable case are conjugacy separable. Martino (2007) [7] using topological results showed that Seifert groups are conjugacy separable. Here we use algebraic method to show that Seifert groups over non-orientable surfaces are conjugacy separable.