Reidemeister zeta functions of low-dimensional almost-crystallographic groups are rational

We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy ?₂ are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.     

The number of Reidemeister moves needed for unknotting

There is a positive constant such that for any diagram representing the unknot, there is a sequence of at most Reidemeister moves that will convert it to a trivial knot diagram, where is the number of crossings in . A similar result holds for elementary moves on a polygonal knot embedded in the 1-skeleton of the interior of a compact, orientable, triangulated 3-manifold . There is a positive constant such that for each , if consists of tetrahedra and is unknotted, then there is a sequence of at most elementary moves in which transforms to a triangle contained inside one tetrahedron of . We obtain explicit values for and .

     

The number of Reidemeister moves for splitting a link

We give an explicit upper bound for the number of Reidemeister moves for deforming a link diagram of a split link to be disconnected.Mathematics Subject Classification (2000): 57M25, 57N10The author is partially supported by Grant-in-Aid for Scientific Research (No. 15740047), Ministry of Education, Science, Sports and Technology, Japan.     

On the category of weak Cayley table morphisms between groups

Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.     

A local-global principle for finiteness properties ofS-arithmetic groups over number fields

In the preceding paper [AT] compactness propertiesC _n andCP _n for locally compact groups were introduced. They generalize the finiteness propertiesF _n andFP _n for discrete groups. In this paper a local-global principle forS-arithmetic groups over number fields is proved. TheS-arithmetic group is of typeF _n , resp.FP _n , if and only if for allp inS thep-adic completionG _p of the corresponding algebraic groupG is of typeC _n resp.CP _n . As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnS-arithmetic subgroup of a semisimple group has all the finiteness propertiesF _n .     

The residual finiteness of positive one-relator groups

It is proven that every positive one-relator group which satisfies the condition has a finite index subgroup which splits as a free product of two free groups amalgamating a finitely generated malnormal subgroup. As a consequence, it is shown that every positive one-relator group is residually finite. It is shown that positive one-relator groups are generically and hence generically residually finite. A new method is given for recognizing malnormal subgroups of free groups. This method employs a 'small cancellation theory' for maps between graphs. Received: August 4, 2000     

Unknotting number and number of Reidemeister moves needed for unlinking

Using unknotting number, we introduce a link diagram invariant of type given in Hass and Nowik (2008) [4], which changes at most by 2 under a Reidemeister move. We show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being splitted with respect to the number of crossings.     

Factor morphisms of nilpotent groups

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 119–125, May–June, 1991.     

Some bounds for the number of coincidences of morphisms between closed riemann surfaces

We give some bounds for the number of coincidences of two morphisms between given compact Riemann surfaces (complete complex algebraic curves) which generalize well known facts about the number of fixed points of automorphisms. In the particular case in which both surfaces are hyperelliptic, our results permit us to obtain a bound for the number of morphisms between them. The proof relies on the idea, first used by Schwarz in the case of automorphisms, of representing a morphism by its action on the set of Weierstrass points.     

Sofic groups and direct finiteness

We construct an analogue of von Neumann's affiliated algebras for sofic group algebras over arbitrary fields. Consequently, we settle Kaplansky's direct finiteness conjecture for sofic groups.     

Smooth skew morphisms of dicyclic groups

Journal of Algebraic Combinatorics - A skew morphism of a finite group A is a permutation $$\varphi $$ on A fixing the identity element of A, and for which there exists an integer-valued function...     

Axioms for the equivariant Lefschetz number and for the Reidemeister trace

We characterize the Reidemeister trace, the equivariant Lefschetz number and the equivariant Reidemeister trace in terms of certain axioms. Dedicated to Albrecht Dold and Edward Fadell     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

Residual finiteness of descending HNN-extension of groups

The paper examines the special case of the general construction of HNN-extensions of groups in which at least one of the associated subgroups is the base group. A criterion is determined for a group obtained in this way to be residually finite. Any group obtained as such an extension from a free nilpotent group of finite rank is residually finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 842–845, June, 1992.