Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group T] (and its companion T) which is an extension of the Ptolemy–Thompson group T by the braid group B1 on infinitely many strands. We prove that T] is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups T] and T are in the same relation with respect to each other as the braid groups BnC1 and Bn, for infinitely many strands n. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable. 20F36, 57M07; 20F38, 20F05, 57N05

In this paper we discuss generic properties of ”random subgroups ” of a given group G. It turns out that in many groups G (even in most exotic of them) the random subgroups have a simple algebraic structure and they ”sit ” inside G in a very particular way. This gives a strong mathematical foundation for cryptanalysis of several group-based cryptosystems and indicates on how to chose ”strong keys”. To illustrate our technique we analyze the Anshel-Anshel-Goldfeld (AAG) cryptosystem and give a mathematical explanation of recent success of some heuristic length-based attacks on it. Furthermore, we design and analyze a new type of attacks, which we term the quotient attacks. Mathematical methods we develop here also indicate how one can try to choose ”parameters ” in AAG to foil the attacks.

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of S3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of S3 branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume. We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. We show that several torsion free 3-manifold groups are not left-orderable. Many of our manifolds are obtained by taking n-fold branched covers along various hyperbolic 2-bridge knots. The paper is organized in the following way: after defining left-orderability we state our main theorem listing branched set links and multiplicity of coverings from which we obtain manifolds with non-left-orderable groups. Then we describe presentations of these groups in a way which allows the

Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's model-theoretic methods, we re-prove exact versions of unprovability results for the Paris-Harrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the Paris-Harrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its model-theoretic techniques and, finally, a model-theoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logic-aware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.

games on braids

The purpose of this paper is the study of the roots in the mapping class groups. Let Σ be a compact oriented surface, possibly with boundary, let P be a finite set of punctures in the interior of Σ, and let M(Σ, P) denote the mapping class group of (Σ, P). We prove that, if Σ is of genus 0, then each f ∈ M(Σ) has at most one m-root for all m ≥ 1. We prove that, if Σ is of genus 1 and has non-empty boundary, then each f ∈ M(Σ) has at most one m-root up to conjugation for all m ≥ 1. We prove that, however, if Σ is of genus ≥ 2, then there exist f, g ∈ M(Σ, P) such that f 2 = g 2, f is not conjugate to g, and none of the conjugates of f commutes with g. Afterwards, we focus our study on the roots of the pseudo-Anosov elements. We prove that, if ∂Σ ̸ = ∅, then each pseudo-Anosov element f ∈ M(Σ, P) has at most one m-root for all m ≥ 1. We prove that, however, if ∂Σ = ∅ and the genus of Σ is ≥ 2, then there exist two pseudo-Anosov elements f, g ∈ M(Σ) (explicitely constructed) such that f m = g m for some m ≥ 2, f is not conjugate to g, and none of the conjugates of f commutes with g. Furthermore, if the genus of Σ is ≡ 0 (mod 4), then we can take m = 2. Finally, we show that, if Γ is a pure subgroup of M(Σ, P) and f ∈ Γ, then f has at most one m-root in Γ for all m ≥ 1. Note that there are finite index pure subgroups in M(Σ, P). AMS Subject Classification: Primary 57M99. Secondary 57N05, 57R30. 1

Abstract. We define a measure of “complexity ” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators ∆ij, which are Garside-like half-twists involving strings i through j, and by counting powered generators ∆k ij as log(|k | + 1) instead of simply |k|. The geometrical complexity is some natural measure of the amount of distortion of the n times punctured disk caused by a homeomorphism. Our main result is that the two notions of complexity are comparable. We also show how to recover a braid from its curve diagram in polynomial time, and we prove that every braid has a σ1-consistent representative of linearly bounded length. The key rôle in the proofs is played by a technique introduced by Agol, Hass, and Thurston.

A left order on a magma (e.g., semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and ̷Lo´s [13] concerning the existence of left orders.

Residual p properties of mapping class groups and surface groups