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.     

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     

Virtual retractions,conjugacy separability and omnipotence

We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.     

Outer automorphism groups of certain 1-relator groups

Grossman first showed that outer automorphism groups of 1-relator groups given by orientable surface groups are residually finite, whence mapping class groups of orientable surfaces are residually finite. Allenby, Kim and Tang showed that outer automorphism groups of cyclically pinched 1-relator groups are residually finite, whence mapping class groups of orientable and non-orientable surfaces are residually finite. In this paper we show that outer automorphism groups of certain conjugacy separable 1-relator groups are residually finite.     

Bianchi groups are conjugacy separable

We prove that non-uniform arithmetic lattices of SL₂(C) and consequently the Bianchi groups are conjugacy separable. The proof is based on recent deep results of Agol, Long, Reid and Minasyan. The conjugacy separability of groups commensurable with limit groups is also established.     

Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions

In this note we shall give characterisations for HNN extensions of non-cyclic polycyclic-by-finite groups with normal infinite cyclic associated subgroups to be residually finite, subgroup separable and conjugacy separable.     

特征标表中零点个数很少的有限可解群

（１）与（２）均曾研究过有某特征标恰在两个共轭类上取非零值的有限群。本文讨论上述情形的另一极端,即每个不可约特征标至多在ｐ个共轭类上取零值的有限可解群,这里ｐ为群阶的最小素因子,通过对极小非交换商群的分析,我们将刻划这类群的结构。     

Nielsen Zeta Function, 3-Manifolds, and Asymptotic Expansions in the Nielsen Theory

We prove that the Nielsen zeta function is either a rational function or a radical of a rational function for orientation-preserving homeomorphisms of closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds. In the case of a pseudo-Anosov homeomorphism of a surface, we find an asymptotics for the number of twisted conjugacy classes or for the number of Nielsen fixed-point classes with norm at most x. Bibliography: 20 titles.     

Small cancellation theory with a weakened small cancellation hypothesis. III. The conjugacy problem

In [1] we introduced the small cancellation condition W(4) and developed the basic theory. In this article we solve the conjugacy problem for groups with a finite W(4) presentation. Incumbent of the Lilian and George Lyttle Career Development Chair in Perpetuity, Established by Mrs. Helen Kimmelman, New York.     

The Geometry of Elation Groups of a Finite Projective Space

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge [7] we establish the number of these elation groups.     

Conjugacy Separability of Descending HNN-Extensions of Finitely Generated Abelian Groups

It is proved that an arbitrary descending HNN-extension of a finitely generated Abelian group is conjugacy separable.     

Finite Groups in Which Each Irreducible Character has at Most Two Zeros

Let G be a finite group, Irr(G) denotes the set of irreducible complex characters of G and gG the conjugacy class of G containing element g. A well-known theorem of Burnside([1,Theorem 3.15]) states that every nonlinear X ∈ Irr(G) has a zero on G, that is, an element x (or a conjugacy class xG) of G with X(x) = 0. So, if the number of zeros of character table is very small, we may expect, the structure of group is heavily restricted. For example, [2, Proposition 2.7] claimes that G is a Frobenius group with a complement of order 2 if each row in charcter table has at most one zero (its proof uses the classification of simple groups). In this note, we characterize the finite group G satisfying the following hypothesis:     

The Conjugacy Problem is Solvable in Free-By-Cyclic Groups

We show that the conjugacy problem is solvable in [finitelygenerated free]-by-cyclic groups, by using a result of O. Maslakovathat one can algorithmically find generating sets for the fixedsubgroups of free group automorphisms, and one of P. Brinkmannthat one can determine whether two cyclic words in a free groupare mapped to each other by some power of a given automorphism.We also solve the power conjugacy problem, and give an algorithmto recognize whether two given elements of a finitely generatedfree group are twisted conjugated to each other with respectto a given automorphism. 2000 Mathematics Subject Classification20F10, 20E05.     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．     

A model with no strongly separable almost disjoint families

We answer a question of Shelah and Steprāns [6] by producing a model of ZFC where there are no strongly separable almost disjoint families. The notion of a strongly separable almost disjoint family is a natural variation on the well known notion of a completely separable almost disjoint family, and is closely related to the metrization problem for countable Fréchet groups.     

On the logistic delay differential equations with piecewise constant argument and quasi-periodic time dependence

In this paper, we discuss the quasi-periodic logistic delay differential equations. As a corollary, we give a more sharp result than that in [G. Seifert, J. Differential Equations 164 (2000) 451-458] for the periodic logistic delay differential equations.     

Seifert circles,braid index and the algebraic crossing number

We introduce new operations reducing the number of Seifert circles in link diagrams of a special type. The operations are similar to one described in [Mem. Amer. Math. Soc. 508 (1993)] and [Math. Proc. Cambridge Philos. Soc. 111 (2) (1992) 273]. We discuss a conjecture about the number of Seifert circles that can be canceled by applying the operation repeatedly. We translate the problem into one belonging to graph theory.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

Foliations transverse to fibers of Seifert manifolds

In this paper we prove the conjecture of Jankins and Neumann [JN2] about rotation numbers of products of circle homeomorphisms, which together with other results of [EHN] and [JN2] (mentioned below) implies that a Seifert manifold admits foliations tranverse to its fibers only if it admits such foliations with a projective transverse structure. 1991Mathematics Subject Classification: Primary 55R05, 57R30, Secondary 47A35, 57M99, 58F11. Partially supported by a Sloan Doctoral Dissertation Fellowship, and by the Technion-Israel Institute of Technology.