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Institut für Mathematik und Informatik

Arbeitsgebiete

  • Topologie, mit einem Schwerpunkt in niederdimensionaler Topologie, 2-Komplexe und 3-Mannigsfaltigkeiten. Ich arbeite an der Whiteheadschen Asphärizitätsvermutung.
  • Algebra, mit einem Schwerpunkt auf Gruppentheorie, insbesondere geometrischer Gruppentheorie. Stichworte sind Hyperbolische Gruppen, Kämmungen, Wort- und Konjugationsproblem.
  • Mathematische und nicht-mathematsiche Software, ein Programm zum Gewichtstest und andere Kleinigkeiten.

Mathematische Veröffentlichungen

  • With J. Harlander: Injective Labeled Oriented Trees are Aspherical. A labeled oriented tree (LOT) is called injective, if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof uses a new relative asphericity test based on a lemma of Stallings. Erscheint in: Mathematische Zeitschrift (2017)
  • With J. Harlander: Aspherical Word Labeled Oriented Graphs and Cyclically Presented Groups. A word labeled oriented graph (WLOG) is an oriented graph G on vertices X={ x_1,... ,x_k}, where each oriented edge is labeled by a word in X^{\pm1}. The classical situation, where each edge label is a single letter in X and the underlying graph is a tree is of central importance in view of Whitehead's Asphericity Conjecture and has been intensely studied. We present a class of of aspherical world labeled oriented graphs. This class can be used to produce highly non-injective aspherical labeled oriented trees and also aspherical cyclically presented groups. Journal of knot theory and its ramifications Vol 24 No 5 (2015)
  • With J. Harlander: On primeness of labeled oriented trees. Knot complements are aspherical. Whether this extends to ribbon disc complements, or, equivalently, to standard 2-complexes of labeled oriented trees, remains unresolved. It is known that prime injective labeled oriented trees are diagragramtically reducible, that is, aspherical in a strong combinatorial sense. We show that arbitrary prime labeled oriented trees need not be DR. We conjecture that all injective labeled oriented trees are aspherical and prove the conjecture under natural conditions. Journal of Knot Theory and its Ramifications, Vol. 21 No. 8 (2012)
  • Some spherical diagrams over labeled oriented trees and graphs. We present some spherical diagrams found by computer search. The diagrams are reduced and the images are 2-complexes over labeled oriented trees and graphs. preprint (2011) SpherDiagEx.pdf
  • With J. Harlander: On distinguishing virtual knot groups from knot groups. We use curvature techniques from geometric group theory to produce examples of virtual knot groups that are not classical knot groups. Journal of Knot Theory and its Ramifications, Vol. 19 No. 5 (2010); pp. 695-704
  • Geometrische Gruppentheorie - Ein Einstieg mit dem Computer. Basiswissen für Studium und Mathematikunterricht; 2. Auflage; vieweg-teubner Verlag, 2010, ISBN 978-3-8348-1038-0, 211 Seiten.
  • On the Complexity of Labelled Oriented Trees. We define a notion of complexity for labeled oriented trees (LOTs) and prove that LOTs of complexity 2 are aspherical. We also present a class of LOTs of higher complexity which is aspherical and study the complexity of torus knots. Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 1, February 2010, pp. 11-18 http://www.ias.ac.in/mathsci/vol120/feb2010/PM00172.PDF
  • The Whitehead-Conjecture - an Overview. These notes are an elaboration of a talk held November 3, 2006 at the ''Metzler Fest'' in honour of Wolfgang Metzler's 65-th birthday at the university of Frankfurt. The aim is to give an overview of results concerning Whitehead's asphericity conjecture. Siberian Electronic Mathematical Reports 4; (2007); S. 440 - 449. http://semr.math.nsc.ru/v4/p440-449.pdf
  • With G. Huck: Spherical Diagrams and Labeled Oriented Trees. We develop a method of proving asphericity for standard 2-complexes. This method is based on a Lemma of Stallings. We reverse the orientation of edges in reduced spherical diagrams contradicting their existence. By an application to labeled oriented graphs we obtain new classes of aspherical ribbon disk complements. Proc. of the Edinburgh Math. Society 137A (2007); pp. 519 - 530
  • Geometrische Gruppentheorie - Ein Einstieg mit dem Computer. Basiswissen für Studium und Mathematikunterricht; vieweg Verlag, 2004, ISBN 3-528-03212-X, 206 Seiten
  • With J. Harlander: Generalized knot complements and some aspherical ribbon disc complements. We generalize some aspects of standard knot-theory to all ribbon-disc complements. We study asphericity of the complement of properly embedded links in certain contractible singular 3-manifolds that should be thought off as replacements of the 3-ball in the classical setting. We apply our results to show asphericity of ribbon-disc complements that admit alternating prime projections on some surface. Knot Theory and its Ramifications 12 (7), 2003; pp. 947 - 962
  • With G. Huck: Aspherical Labelled Oriented Trees and Knots. The question of whether ribbon-disc complements, or equivalently standard 2-complexes over labelled oriented trees, are aspherical is of great importance for Whitehead's asphericity conjecture and for an algebraic proof of the asphericity of knot complements. We present here two classes of diagrammatically reducible labelled oriented trees. One of them implies as a corollary a combinatorial proof of the diagrammatic asphericity of spines of the complements of alternating knots. Proc. of the Edinburgh Math. Society (44) 2001; pages 285-294 Proposition 5.1 von diesem Aufsatz ist hier nochmal separat in Deutsch aufgeschrieben in seiner reinen graphentheoretischen Form. BaumGraph.pdf.
  • With G. Huck: Cancellation Diagrams with non-positive Curvature. We present four generalised small cancellation conditions for finite presentations and solve the word- and conjugacy problem in each case. Our conditions W and W* contain the non-metric small cancellation cases C(6), C(4)T(4), C(3)T(6) but are considerably more general. W also contains as a special case the small cancellation condition W(6) of Juhasz. If a finite presentation satisfies W or W* then it has a quadratic isoperimetric inequality and therefore solvable word problem. For the class W this was first observed by Gersten. Our main result here is the proof of the conjugacy problem for the classes W* and W which uses the geometry of non-positively curved piecewise Euclidean complexes developed by Bridson. The conditions V and V* generalise the small cancellation conditions C(7), C(5)T(4), C(4)T(5), C(3)T(7). If a finite presentation satisfies the condition V or V*, then it has a linear isoperimetric inequality and hence the group is hyperbolic. Computational and Geometric Aspects of Modern Algebra; Michael Atkinson et al. Editors; London Math. Soc. Lec. Notes Ser. 275 (2000); pages 128-149
  • Some aspherical labeled oriented graphs. We use Klyachko's result on cell decompositions of the 2-sphere to show the asphericity of certain classes of labeled oriented trees and, more general of some labeled oriented graphs. The author is supported by INTAS, Project 97-808. Low-Dimensional Topology and Combinatorial Group Theory; Proceedings of the International Conference, Kiev; 2000; Editor: S. Matveev; pages 307-314
  • With A. Christensen: On the Impossibility of a Generalisation of the HOMFLY - Polynomial to Labelled Oriented Graphs. We show that any generalisation of the HOMFLY polynomial to labelled oriented graphs is almost trivial. This means on the one hand that the geometry of links is essential for computing this polynomial. On the other hand it shows that this polynomial cannot serve as an invariant for LOG groups. Annales de la Faculte des Sciences de Toulouse 5 (3); 1996; pp. 407-419
  • With G. Huck: Weight tests and hyperbolic groups. Hier wird unter anderem der Gewichtstest definiert. The notion of a reduced diagram plays a fundamental role in small cancellation theory and in tests for detecting asphericity of 2-complexes. By introducing vertex reduced as a stricter form of reducedness in diagrams we obtain a new combinatorial notion of asphericity for 2-complexes, called vertex asphericity, which generalises diagrammatic reducibility and implies diagrammatic asphericity. This leads to a generalisation and simplification in applying the weight test and the cycle test to detect asphericity of 2-complexes and (for the hyperbolic versions of these tests) to detect hyperbolic group presentations. In the end, we present an application to labelled oriented graphs. Combinatorial and Geometric Group Theory; Cambridge University Press; London Math. Soc. Lecture Note Ser. 204; Editors: A. Duncan, N. Gilbert, J. Howie; 1995; pp. 174-183
  • On the Realization of Wirtinger Presentations as Knot Groups. The weight test of Gersten and Pride is modified, to test whether a given abstract Wirtinger presentation occurs as the geometric Wirtinger presentation of a knot. A more refined version of this test then shows that the groups presented by certain abstract Wirtinger presentations are hyperbolic and so do not occur as knot groups. We give an infinite class of examples of such groups with symmetric Alexander polynomial. Journal of knot theory and its Ramifications; Volume 3; 1994; pp. 211-222
  • With G. Huck: A Bicombing that implies a Sub - Exponential Isoperimetric Inequality. The idea of applying isoperimetric functions to group theory is due to M. Gromov. We introduce the concept of a bicombing of narrow shape which generalises the usual notion of bicombing. Our bicombing is related to but different from the combings defined by M. Bridson. If the Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub - exponential isoperimetric inequality, as well as a polynomial isodiametric inequality. We give an infinite class of examples which are not bicombable in the usual sense but admit bicombings of narrow shape. Proceedings of the Edinburgh Math. Soc. 36; 1993; pp. 515-523
  • With G. Huck: Applications of diagrams to decision problems. Classical decision problems such as the word- and conjugacy problem are introduced and methods are given for solving them in certain cases. All the methods we present involve Van-Kampen diagrams as one of the most powerful tools when dealing with the classical decision problems. Two-dimensional homotopy and combinatorial group theory; Cambridge University Press; London Math. Soc. Lecture Note Ser.; Editors: C. Hog-Angeloni, W. Metzler, A. Sieradski; 1993; pp. 189-218
  • Mit G. Huck: Ein verallgemeinerter Gewichtstest mit Anwendungen auf Baumpräsentationen. Mathematische Zeitschrift 211; 1992; pp. 351-367
  • A reduced Spherical Diagram into a Ribbon-Disk Complement and related Examples. The Whitehead Conjecture states, that subcomplexes of aspherical 2-complexes are aspherical. Howie pointed out that ''most of it'' would be proved, if the asphericity of ribbon-disk complements were shown, but each non-aspherical ribbon- disk complement is already a counterexample to the Whitehead conjecture. In the last years different notions of asphericity such as diagrammatically reducible (DR) or diagrammatically aspherical (DA) were defined for 2-complexes and became more and more important for instance in the theory of equations over groups. In this paper I will exhibit examples of non DR and non DA ribbon-disk complements, which answers in the negative a question raised by S. Gersten. Furthermore the existence of combinatorial reduced surface mappings into ribbon-disk complements is proved. Topology and Combinatorial Group Theory; Springer Lecture Notes in Math. 1440; Editor P. Latiolais; 1990; pp. 175-185