N. D. Gilbert

Presentations of the inverse braid monoid

The inverse braid monoid consists of n-string braids from which some subset of the strings has been deleted. Such partial braids may be composed by concatenation followed by the deletion of any incomplete strings, and this operation gives the structure of an inverse monoid. We derive presentations of by first using a groupoid presentation, and then deriving equivalent inverse monoid presentations. In particular, we offer a new derivation of the presentation for that has been given by Easdown and Lavers.

Monoid presentations and associated groupoids

We consider properties of a 2-complex associated by Squier to a monoid presentation. We show that the fundamental groupoid admits a monoid structure, and we establish a relationship between its group completion and the fundamental group of the 2-complex. We also treat a modified complex, due to Pride, for monoid presentations of groups, and compute the structure of the fundamental groupoid in this setting.

Combinatorial and geometric group theory : Edinburgh, 1993

On bounded languages and the geometry of nilpotent groups M. R. Bridson and R. H. Gilman Finitely presented groups and the finite generation of exterior powers C. J. B. Brookes Semigroup presentations and minimal ideals C. M. Campbell, E. F. Robertson, N. Ruskuc and R. M. Thomas Generalized trees and Lambda-trees I. M. Chiswell The mathematician who had little wisdom: a story and some mathematics D. E. Cohen Palindromic automorphisms of free groups D. J. Collins A freiheitssatz for certain one-relator amalgamated products B. Fine, F. Roehl and G. Rosenberger Isoperimetric functions of groups and exotic cohomology S. M. Gersten Some embedding theorems and undecidability questions for groups C. M. Gordon Some results on bounded cohomology R. I. Grigorchuk On perfect subgroups of one-relator groups J. Harlander Weight tests and hyperbolic groups G. Huck and S. Rosebrock A non-residually finite, relatively finitely presented group in the variety N2A O. G. Kharlampovich and M. V. Sapir Hierarchical decompositions, generalized Tate cohomology and groups of type FP P. H. Kropholler Tree-lattices and lattices in Lie groups A. Lubotzky Generalizations of Fibonacci numbers, groups and manifolds C. Maclachlan Knotted surfaces in the 4-sphere with no minimal Seifert manifolds T. Maeda The higher geometric invariants of modules over Noetherian group rings H. Meinert On calculation of width in free groups A. Yu Olshanskii Hilbert modular groups and isoperimetric inequalities C. Pittet On systems of equations in free groups A. A. Razborov Cogrowth and essentiality in groups and algebras A. Rosenmann Regular geodesic languages for 2-step nilpotent groups M. Stoll Finding indivisible Nielsen paths for a train track map E. C. Turner More on Burnsides problem E. Zelmanov Problem session.

Computational and geometric aspects of modern algebra

Forword Participants 1. Lie methods in growth of groups and groups of finite width Laurent Bartholdi and Rostislav I. Grigorchuk 2. Translation numbers of groups acting on quasiconvex spaces Gregory R. Conner 3. On a term rewriting system controlled by sequences of integers Ales Drapal 4. On certain finite generalized tetrahedron groups M. Edjvet, G. Rosenberger, M. Stille and R. M. Thomas 5. Efficient computation in word-hyperbolic groups David B. A. Epstein and Derek F. Holt 6. Constructing hyperbolic manifolds B. Everitt and C. Maclachlan 7. Computing in groups with exponent six George Havas, M. F. Newman, Alice C. Niemeyer and Charles C. Sims 8. Rewriting as a special case of non-commutative Grobner basis theory Anne Heyworth 9. Detecting 3-manifold presentations Cynthia Hog-Angeloni 10. In search of a word with special combinatorial properties Stepan Holub 11. Cancellation diagrams with non-positive curvature Gunther Huck and Stephan Rosebrock 12. Some applications of prefix-rewriting in monoids, groups and rings Klaus Madlener and Friedrich Otto 13. Verallgemeinerte biasinvarianten und ihre berechnung Wolfgang Metzler 14. On groups which act freely and properly on finite dimensional homotopy spheres Guido Mislin and Olympia Talelli 15. On confinal dynamics of rooted tree automorphisms V. V. Nekrashevych and V. I. Suchansky 16. An asymptotic invariant of surface groups Amnon Rosenmann 17. A cutpoint tree for a continuum Eric L. Swenson 18. Generalised triangle groups of type (2, m, 2) Alun G. T. Williams.

Weight tests and hyperbolic groups

The notion of 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 generalizes diagrammatic reducibility and implies diagrammatic asphericity. This leads to a generalization and simpliication in applying the weight test 2] and the cycle test 6] 7] 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 labeled oriented graphs. We would like to thank the referee for his helpful suggestions. 1 Basic Deenitions A p.l. map between 2-complexes is called combinatorial, if each open cell is mapped home-omorphically onto its image. A 2-dimensional nite CW-complex is called combinatorial, if the attaching maps of the 2-cells are combinatorial relative to a suitable polygonal subdivision of their boundary. Let K P be the standard 2-complex of the presentation P (we assume all presentations to be nite). A diagram is a combinatorial map f: M ! K P , where M is a combinatorial subcomplex of an orientable 2-manifold. A spherical diagram is a diagram f: S ! K P , where S is the 2-sphere. These deenitions may be found for example in 1], 2], 6], 7] or 8]. The Whitehead graph W P of K P is the boundary of a regular neighborhood of the only vertex of K P (see 6] or 7]). It consists of two vertices +x i and ?x i for each generator x i of P which correspond to the beginning and the end of the edge labeled x i in K P. The edges of W P are the corners of the 2-cells of the 2-complex. The star graph S P is the same as the Whitehead graph if no relator of P is a proper power. Let F denote the free group on the generators of P. If a relator R i of P has the form w k i i with w i not a proper power, then the star graph S P is the Whitehead graph of the presentation < x 1 denote by d(R i) the length of R i , that is the sum of the absolute values of the exponents of R i. For a 2-cell …

Identities between sets of relations

Abstract Algebraic structure theorems for relative and triad homotopy groups of a cellular model of a group presentation are used to study notions of dependence between the relations.

Fibrations of Ordered Groupoids and the Factorization of Ordered Functors

We investigate canonical factorizations of ordered functors of ordered groupoids through star-surjective functors. Our main construction is a quotient ordered groupoid, depending on an ordered version of the notion of normal subgroupoid, that results in the factorization of an ordered functor as a star-surjective functor followed by a star-injective functor. Any star-injective functor possesses a universal factorization through a covering, by Ehresmann’s Maximum Enlargement Theorem. We also show that any ordered functor has a canonical factorization through a functor with the ordered homotopy lifting property.

The idempotent problem for an inverse monoid

We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid are related to the formal language properties of its idempotent problem. We show that if an inverse monoid is either E-unitary or has a finite set of idempotents, then its idempotent problem is regular if and only if the inverse monoid is finite. We also give examples of inverse monoids with context-free idempotent problems, including all Bruck–Reilly extensions of finite groups.

The Graph Expansion of an Ordered Groupoid

We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlisters P-theorem for incompressible ordered groupoids.

HNN extensions of inverse semigroups with zero

We study a construction of an HNN extension for inverse semigroups with zero. We prove a normal form for the elements of the universal group of an inverse semigroup that is categorical at zero, and use it to establish structural results for the universal group of an HNN extension. Our main application of the HNN construction is to show that graph inverse semigroups – including the polycyclic monoids – admit HNN decompositions in a natural way, and that this leads to concise presentations for them.