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14
The braided PtolemyThompson group is finitely presented
, 2008
"... 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 con ..."
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Cited by 19 (9 self)
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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.
On the complexity of braids
, 2004
"... 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 Garsidelike halftwists involving strings i through j, an ..."
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Cited by 15 (3 self)
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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 Garsidelike halftwists 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 σ1consistent representative of linearly bounded length. The key rôle in the proofs is played by a technique introduced by Agol, Hass, and Thurston.
Nonleftorderable 3manifold groups
 Canadian Math. Bull
"... We show that several torsion free 3manifold groups are not leftorderable. 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 ..."
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Cited by 13 (4 self)
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We show that several torsion free 3manifold groups are not leftorderable. 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 leftorderable. Many other examples of nonorderable groups are obtained by taking 3fold branched covers of S3 branched along various hyperbolic 2bridge knots. The manifold obtained in such a way from the 52 knot is of special interest as it is conjectured to be the hyperbolic 3manifold with the smallest volume. We investigate the orderability properties of fundamental groups of 3dimensional manifolds. We show that several torsion free 3manifold groups are not leftorderable. Many of our manifolds are obtained by taking nfold branched covers along various hyperbolic 2bridge knots. The paper is organized in the following way: after defining leftorderability we state our main theorem listing branched set links and multiplicity of coverings from which we obtain manifolds with nonleftorderable groups. Then we describe presentations of these groups in a way which allows the
Random subgroups and analysis of the lengthbased and quotient attacks
 Journal of Mathematical Cryptology
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Compactness of the space of left orders
, 2007
"... 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 nonassociati ..."
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Cited by 5 (0 self)
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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.
Roots in the mapping class groups
, 2008
"... 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 eac ..."
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Cited by 5 (0 self)
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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 mroot for all m ≥ 1. We prove that, if Σ is of genus 1 and has nonempty boundary, then each f ∈ M(Σ) has at most one mroot 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 pseudoAnosov elements. We prove that, if ∂Σ ̸ = ∅, then each pseudoAnosov element f ∈ M(Σ, P) has at most one mroot for all m ≥ 1. We prove that, however, if ∂Σ = ∅ and the genus of Σ is ≥ 2, then there exist two pseudoAnosov 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 mroot 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
Spaces of orders and their Turing degree spectra
 SUBMITTED TO ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, whichis leftinvariant under the group operation. Right orders and biorders are defined similarly. In particular, we ..."
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Cited by 3 (0 self)
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We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, whichis leftinvariant under the group operation. Right orders and biorders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s investigation of orders on groups in topology [28] and Solomon’s investigation of these orders in computable algebra [31]. Furthermore, we establish that a computable free group Fn of rank n>1 has a biorder in every Turing degree.
PROBLEMS RELATED TO ARTIN GROUPS
"... Artin groups span a wide range of groups from braid groups to free groups to free abelian groups, as well as many more exotic groups. In recent years, Artin groups and their subgroups have proved to be a rich source of examples and counterexamples of interesting phenomena in geometry and group theor ..."
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Artin groups span a wide range of groups from braid groups to free groups to free abelian groups, as well as many more exotic groups. In recent years, Artin groups and their subgroups have proved to be a rich source of examples and counterexamples of interesting phenomena in geometry and group theory. Artin groups, like Coxeter groups, are defined by presentations of a particular form. A Coxeter graph is a finite, simplicial graph Γ with vertex set S and edges labeled by integers m ≥ 2. We denote by m(s, t) the label on the edge connecting vertices s and t. By convention, m(s, t) = ∞ if s, t are not connected by an edge. The Artin group associated to a Coxeter graph Γ is the group A given by the presentation A = 〈S  sts... ︸ ︷ ︷ ︸ m(s,t) = tst... ︸ ︷ ︷ ︸ m(s,t) if s, t are connected by an edge〉. We call Γ the defining graph for A. Adding additional relations s2 = 1 for all s ∈ S gives