A strong geometric hyperbolicity property for directed graphs and monoids

We introduce and study a strong "thin triangle"' condition for directed graphs, which generalises the usual notion of hyperbolicity for a metric space. We prove that finitely generated left cancellative monoids whose right Cayley graphs satisfy this condition must be finitely presented with polynomial Dehn functions, and hence word problems in NP. Under the additional assumption of right cancellativity (or in some cases the weaker condition of bounded indegree), they also admit algorithms for more fundamentally semigroup-theoretic decision problems such as Green's relations L, R, J, D and the corresponding pre-orders. In contrast, we exhibit a right cancellative (but not left cancellative) finitely generated monoid (in fact, an infinite class of them) whose Cayley graph is a essentially a tree (hence hyperbolic in our sense and probably any reasonable sense), but which is not even recursively presentable. This seems to be strong evidence that no geometric notion of hyperbolicity will be strong enough to yield much information about finitely generated monoids in absolute generality.Comment: Exposition improved. Results unchange

Monoids in the mapping class group

In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.Comment: 36 pages, 18 figure

Stallings graphs for quasi-convex subgroups

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the conjugacy or the almost malnormality problems. Our results extend earlier algorithmic results for the more restricted class of virtually free groups. We also extend our construction to relatively quasi-convex subgroups of relatively hyperbolic groups, under certain additional conditions.Comment: 40 pages. New and improved versio

Asymptotic invariants, complexity of groups and related problems

We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.Comment: 86 pages. Preliminary version, comments are welcome. v2: some references added, misprints fixed, some changes suggested by the readers are made. 88 pages. v3: more readers' suggestions implemented, index added, the list of references improved. This version is submitted to a journal. v4: The paper is accepted in Bulletin of Mathematical Science

Braids: A Survey

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labelled, as we encounter them.Comment: Final version, revised to take account of the comments of readers. A review article, to appear in the Handbook of Knot Theory, edited by W. Menasco and M. Thistlethwaite. 91 pages, 24 figure

Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2

The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time

Laminations and groups of homeomorphisms of the circle

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that pi_1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so pi_1(M) is isomorphic to a subgroup of Homeo(S^1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S^1. As a corollary, the Weeks manifold does not admit a tight essential lamination, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston's universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem.Comment: 50 pages, 12 figures. Ver 2: minor improvement

Polygraphs: From Rewriting to Higher Categories

Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the abstract point of view offered by homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. It is meant to be progressive, with little requirements on the background of the reader, apart from basic category theory, and is illustrated with algorithmic computations on algebraic structures. The second half introduces and studies the general notion of n-polygraph, dealing with the homotopy theory of those. It constructs the folk model structure on the category of strict higher categories and exhibits polygraphs as cofibrant objects. This allows extending to higher dimensional structures the coherence results developed in the first half