Susan Hermiller

Artin groups, rewriting systems and three-manifolds

Abstract We construct finite complete rewriting systems for two large classes of Artin groups: those of finite type, and those whose defining graphs are based on trees. The constructions in the two cases are quite different; while the construction for Artin groups of finite type uses normal forms introduced through work on complex hyperplane arrangements, the rewriting systems for Artin groups based on trees are constructed via three-manifold topology. This construction naturally leads to the question: Which Artin groups are three-manifold groups? Although we do not have a complete solution, the answer, it seems, is “not many”.

Rewriting systems for Coxeter groups

Abstract A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this paper it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting system, using the usual generators a 1 ,…, a g , b 1 ,…, b g along with generators representing their inverses. Constructions of finite complete rewriting systems are also given for any Coxeter group G satisfying one of the following hypotheses. (1) G has three or fewer generators. (2) G does not contain a special subgroup of the form 〈 s i , s j , s k | s 2 i = s 2 j = s 2 k = ( s i s j ) 2 = ( s i s k ) m = ( s j s k ) n = 〈 with m and n both finite and not both equal to two.

Tame combings, almost convexity and rewriting systems for groups

A finite complete rewriting system for a group is a finite presentation which gives an algorithmic solution to the word problem. Finite complete rewriting systems have proven to be useful in geometric group theory, yet little is known about the geometry of groups admitting such rewriting systems. Here we indicate some of the geometry that is implicit in groups with various types of finite rewriting systems. For example, we show that any group admitting a finite complete rewriting system is tame 1-combable, and if the rewriting system is geodesic, then the group is almost convex. Several properties for finitely presented groups have been defined which can be used to show that a closed P2-irreducible three-manifold has universal cover homeomorphic to R3. For example, work of Poenaru [P] shows that if the fundamental group is infinite and satisfies Cannon’s almost convexity property, then the universal cover is simply connected at infinity, and hence is R3 (see [BT]). Casson later discovered the property C2, which regrettably is presentation dependent, but which also implies that the universal cover is R3 [S-G]. Brick and Mihalik generalized the condition C2 to the quasi-simply-filtered condition [B-M], which is independent of presentation and also implies the covering property. Later Mihalik and Tschantz [M-T] defined the notion of a tame 1-combing for a finitely presented group, which implies the quasi-simply-filtered condition, and showed that asynchronously automatic groups and semihyperbolic groups are tame 1-combable. Using a fairly geometric argument we show that groups with finite complete rewriting systems have tame 1-combings. Using similar techniques, we obtain

Electronic structure of polyhedral alkanes

Self‐consistent‐field wave functions have been obtained for tetrahedrane, cubane, and dodecahedrane with a flexible s, p basis set, (9s5p/4s), [4s3p/2s]. The CC and CH distances were optimized to 0.001 A. Ionization potentials have been computed both by the ΔSCF and the Koopmans’ Theorem methods. The full spatial symmetry was used in the calculations, and some of the required considerations for dodecahedrane are discussed.

Measuring the tameness of almost convex groups

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points x in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness flnctions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions.

Minimal almost convexity

Abstract In this article we show that the Baumslag–Solitar group BS(1, 2) is minimally almost convex, or MAC. We also show that BS(1, 2) does not satisfy Poénaru’s almost convexity condition P(2), and hence the condition P(2) is strictly stronger than MAC. Finally, we show that the groups BS(1, q) for q ≥ 7 and Stallings’ non-FP3 group do not satisfy MAC. As a consequence, the condition MAC is not a commensurability invariant.

Conjugacy growth series and languages in groups

In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products. 2010 Mathematics Subject Classification: 20F65, 20E45.

Rewriting Systems and Geometric Three-Manifolds

The fundamental groups of most (conjecturally, all) closed three-manifolds with uniform geometries have finite complete rewriting systems. The fundamental groups of a large class of amalgams of circle bundles also have finite complete rewriting systems. The general case remains open.

GROUPS WHOSE GEODESICS ARE LOCALLY TESTABLE

A regular set of words is (k-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length k. In this article we study groups for which the set of all geodesic words with respect to some generating set is (k-)locally testable, and we call such groups (k-)locally testable. We show that a group is 1-locally testable if and only if it is free abelian. We show that the class of (k-)locally testable groups is closed under taking finite direct products. We show also that a locally testable group has finitely many conjugacy classes of torsion elements. Our work involved computer investigations of specific groups, for which purpose we implemented an algorithm in GAP to compute a finite state automaton with language equal to the set of all geodesics of a group (assuming that this language is regular), starting from a shortlex automatic structure. We provide a brief description of that algorithm.

Isoperimetric Inequalities for Soluble Groups

We approach the question of which soluble groups are automatic. We describe a class of nilpotent-by-Abelian groups which need to be studied in order to answer this question. We show that the nilpotent-by-cyclic groups in this class have exponential isoperimetric inequality and so cannot be automatic.