A Hybrid Search Algorithm for the Whitehead Minimization Problem

The Whitehead Minimization problem is a problem of finding elements of the minimal length in the automorphic orbit of a given element of a free group. The classical algorithm of Whitehead that solves the problem depends exponentially on the group rank. Moreover, it can be easily shown that exponential blowout occurs when a word of minimal length has been reached and, therefore, is inevitable except for some trivial cases. In this paper we introduce a deterministic Hybrid search algorithm and its stochastic variation for solving the Whitehead minimization problem. Both algorithms use search heuristics that allow one to find a length-reducing automorphism in polynomial time on most inputs and significantly improve the reduction procedure. The stochastic version of the algorithm employs a probabilistic system that decides in polynomial time whether or not a word is minimal. The stochastic algorithm is very robust. It has never happened that a non-minimal element has been claimed to be minimal

Generators of split extensions of Abelian groups by cyclic groups

Let $G \simeq M \rtimes C$ be an $n$-generator group with $M$ Abelian and $C$ cyclic. We study the Nielsen equivalence classes and T-systems of generating $n$-tuples of $G$. The subgroup $M$ can be turned into a finitely generated faithful module over a suitable quotient $R$ of the integral group ring of $C$. When $C$ is infinite, we show that the Nielsen equivalence classes of the generating $n$-tuples of $G$ correspond bijectively to the orbits of unimodular rows in $M^{n -1}$ under the action of a subgroup of $GL_{n - 1}(R)$. Making no assumption on the cardinality of $C$, we exhibit a complete invariant of Nielsen equivalence in the case $M \simeq R$. As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, lamplighter groups and split metacyclic groups.Comment: 36 pages, The former Theorem F.ii has been retracted because the proof was wrong and couldn't be repaired. To appear in Groups, Geometry and Dynamic

A characterization of virtually embedded subsurfaces in 3-manifolds

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.Comment: 28 pages. Errors of previous Proposition 3.1 and Formula 7.2 correcte

Algorithmic problems for free-abelian times free groups

We study direct products of free-abelian and free groups with special emphasis on algorithmic problems. After giving natural extensions of standard notions into that family, we find an explicit expression for an arbitrary endomorphism of \ZZ^m \times F_n. These tools are used to solve several algorithmic and decision problems for \ZZ^m \times F_n : the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.Comment: 38 page

Utilising IGV approach to identify factors affecting web usability

Due to the changing nature of Internet technology and user needs,continuous web evaluation has become very important in determining the usability of web sites. However, web designers often face problems in identifying the right criteria for evaluation. Despite the growing number of guidelines and other literature on web design and evaluation,each of them varies in terms of quality, coverage, relevancy, and suitability. With this in mind, a study using IGV approach was carried out to identify key generic criteria that need to be taken into consideration by designers or others when assessing the overall usability of web sites. The results of the study include a comprehensive list of the identified usability criteria that were grouped into 7 major factors - screen appearance, content, accessibility, navigation, media use, interactivity, and consistenc