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Journal articles on the topic 'Garside groups'

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Published: 4 June 2021

Last updated: 10 February 2022

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1

Arcis, Diego, and Luis Paris. "Ordering Garside groups." International Journal of Algebra and Computation 29, no. 05 (July 8, 2019): 861–83. http://dx.doi.org/10.1142/s0218196719500322.

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We introduce a structure on a Garside group that we call Dehornoy structure and we show that an iteration of such a structure leads to a left-order on the group. We define two conditions on a Garside group [Formula: see text] and we show that if [Formula: see text] satisfies these two conditions, then [Formula: see text] has a Dehornoy structure. Then, we show that the Artin groups of type [Formula: see text] and of type [Formula: see text], [Formula: see text] satisfy these conditions, and therefore have Dehornoy structures. As indicated by the terminology, one of the orders obtained by this method on the Artin groups of type [Formula: see text] coincides with the Dehornoy order.

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2

Lee, Eon-Kyung, and Sang-Jin Lee. "Periodic elements in Garside groups." Journal of Pure and Applied Algebra 215, no. 10 (October 2011): 2295–314. http://dx.doi.org/10.1016/j.jpaa.2010.12.011.

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3

Chouraqui, Fabienne. "Left orders in Garside groups." International Journal of Algebra and Computation 26, no. 07 (November 2016): 1349–59. http://dx.doi.org/10.1142/s0218196716500570.

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We consider the structure group of a non-degenerate symmetric (non-trivial) set-theoretical solution of the quantum Yang–Baxter equation. This is a Bieberbach group and also a Garside group. We show this group is not bi-orderable, that is it does not admit a total order which is invariant under left and right multiplications. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with a recurrent left order and with space of left orders homeomorphic to the Cantor set, while there exist others that are even not unique product groups.

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4

Lee, Eon-Kyung, and Sang Jin Lee. "Abelian Subgroups of Garside Groups." Communications in Algebra 36, no. 3 (March 7, 2008): 1121–39. http://dx.doi.org/10.1080/00927870701715605.

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5

Calvez, Matthieu, and Bert Wiest. "Curve graphs and Garside groups." Geometriae Dedicata 188, no. 1 (November 28, 2016): 195–213. http://dx.doi.org/10.1007/s10711-016-0213-x.

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6

Godelle, Eddy. "Parabolic subgroups of Garside groups." Journal of Algebra 317, no. 1 (November 2007): 1–16. http://dx.doi.org/10.1016/j.jalgebra.2007.05.024.

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7

Franco, Nuno, and Juan González-Meneses. "Conjugacy problem for braid groups and Garside groups." Journal of Algebra 266, no. 1 (August 2003): 112–32. http://dx.doi.org/10.1016/s0021-8693(03)00292-8.

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8

Sibert, Herve´. "EXTRACTION OF ROOTS IN GARSIDE GROUPS." Communications in Algebra 30, no. 6 (June 19, 2002): 2915–27. http://dx.doi.org/10.1081/agb-120003997.

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9

Chouraqui, Fabienne. "Garside Groups and Yang–Baxter Equation." Communications in Algebra 38, no. 12 (December 15, 2010): 4441–60. http://dx.doi.org/10.1080/00927870903386502.

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10

Lee, Sang Jin. "Garside groups are strongly translation discrete." Journal of Algebra 309, no. 2 (March 2007): 594–609. http://dx.doi.org/10.1016/j.jalgebra.2006.03.018.

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11

Dehornoy, Patrick, and Luis Paris. "Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups." Proceedings of the London Mathematical Society 79, no. 3 (November 1999): 569–604. http://dx.doi.org/10.1112/s0024611599012071.

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12

Godelle, Eddy. "Parabolic subgroups of Garside groups II: Ribbons." Journal of Pure and Applied Algebra 214, no. 11 (November 2010): 2044–62. http://dx.doi.org/10.1016/j.jpaa.2010.02.010.

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13

Birman, Joan S., Volker Gebhardt, and Juan González-Meneses. "Conjugacy in Garside groups III: Periodic braids." Journal of Algebra 316, no. 2 (October 2007): 746–76. http://dx.doi.org/10.1016/j.jalgebra.2007.02.002.

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14

Charney, Ruth, and John Meier. "The language of geodesics for Garside groups." Mathematische Zeitschrift 248, no. 3 (May 5, 2004): 495–509. http://dx.doi.org/10.1007/s00209-004-0666-8.

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15

Gebhardt, Volker, and Juan González-Meneses. "The cyclic sliding operation in Garside groups." Mathematische Zeitschrift 265, no. 1 (March 26, 2009): 85–114. http://dx.doi.org/10.1007/s00209-009-0502-2.

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16

Huang, Jingyin, and Damian Osajda. "Helly meets Garside and Artin." Inventiones mathematicae 225, no. 2 (February 15, 2021): 395–426. http://dx.doi.org/10.1007/s00222-021-01030-8.

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AbstractA graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

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17

Antolín, Yago, and Luis Paris. "Transverse properties of parabolic subgroups of Garside groups." Israel Journal of Mathematics 241, no. 2 (February 16, 2021): 501–26. http://dx.doi.org/10.1007/s11856-021-2100-x.

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18

GODELLE, EDDY, and LUIS PARIS. "PREGARSIDE MONOIDS AND GROUPS, PARABOLICITY, AMALGAMATION, AND FC PROPERTY." International Journal of Algebra and Computation 23, no. 06 (September 2013): 1431–67. http://dx.doi.org/10.1142/s021819671350029x.

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We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin–Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parabolic subgroups in terms of these presentations. Afterwards we prove that the amalgamated product of two preGarside groups along a common parabolic subgroup is again a preGarside group. This enables us to define the family of preGarside groups of FC type as the smallest family of preGarside groups that contains the Garside groups and that is closed by amalgamation along parabolic subgroups. Finally, we make an algebraic and combinatorial study on FC type preGarside groups and their parabolic subgroups.

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19

Schleimer, Saul, and Bert Wiest. "Garside theory and subsurfaces: Some examples in braid groups." Groups Complexity Cryptology 11, no. 2 (November 1, 2019): 61–75. http://dx.doi.org/10.1515/gcc-2019-2007.

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Abstract Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with N strands and of Garside length L, the sliding circuit set should have at most {C\cdot L^{N-2}} elements, for some constant C. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are “almost reducible” in multiple ways, and act on the curve graph with small translation distance.

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20

CORNWELL, CHRISTOPHER R., and STEPHEN P. HUMPHRIES. "COUNTING FUNDAMENTAL PATHS IN CERTAIN GARSIDE SEMIGROUPS." Journal of Knot Theory and Its Ramifications 17, no. 02 (February 2008): 191–211. http://dx.doi.org/10.1142/s0218216508006051.

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For elements a, b of a monoid, define the word pk(a,b) = abab⋯ of length k. We find the number of words in a, b which are equal to pk(a,b)n in the Artin semigroup < a,b|pk(a,b) = pk(b,a) >. This number is related to counting certain paths in the ℕ × ℕ lattice. These Artin groups are examples of two generator Garside groups. We also define other examples of Garside groups G on more than two generators, having fundamental word Δ, and similarly find the number of words equal in G to Δn.

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21

Zheng, Hao. "A New Approach to Extracting Roots in Garside Groups." Communications in Algebra 34, no. 5 (June 2006): 1793–802. http://dx.doi.org/10.1080/00927870500542762.

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22

Rump, Wolfgang. "Decomposition of Garside groups and self-similar L-algebras." Journal of Algebra 485 (September 2017): 118–41. http://dx.doi.org/10.1016/j.jalgebra.2017.04.023.

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23

Hohlweg, Christophe, Philippe Nadeau, and Nathan Williams. "Automata, reduced words and Garside shadows in Coxeter groups." Journal of Algebra 457 (July 2016): 431–56. http://dx.doi.org/10.1016/j.jalgebra.2016.04.006.

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24

Picantin, Matthieu. "Automatic Structures for Torus Link Groups." Journal of Knot Theory and Its Ramifications 12, no. 06 (September 2003): 833–66. http://dx.doi.org/10.1142/s0218216503002627.

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A general result of Epstein and Thurston implies that all link groups are automatic, but the proof provides no explicit automaton. Here we show that the groups of all torus links are groups of fractions of so-called Garside monoids, i.e., roughly speaking, monoids with a good theory of divisibility, which allows us to reprove that those groups are automatic, but, in addition, gives a completely explicit description of the involved automata, thus partially answering a question of D. F. Holt.

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25

DEHORNOY, PATRICK, and BERT WIEST. "ON WORD REVERSING IN BRAID GROUPS." International Journal of Algebra and Computation 16, no. 05 (October 2006): 941–57. http://dx.doi.org/10.1142/s021819670600327x.

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It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear.

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26

Gebhardt, Volker, and Juan González-Meneses. "Solving the conjugacy problem in Garside groups by cyclic sliding." Journal of Symbolic Computation 45, no. 6 (June 2010): 629–56. http://dx.doi.org/10.1016/j.jsc.2010.01.013.

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27

Charney, R., J. Meier, and K. Whittlesey. "Bestvina's Normal Form Complex and the Homology of Garside Groups." Geometriae Dedicata 105, no. 1 (April 2004): 171–88. http://dx.doi.org/10.1023/b:geom.0000024696.69357.73.

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28

Rump, Wolfgang. "Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups." Forum Mathematicum 30, no. 4 (July 1, 2018): 973–95. http://dx.doi.org/10.1515/forum-2017-0108.

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AbstractIt is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular latticeX, admits a natural embedding into a group{G(X)}with a lattice ordering so that{G(X)}determinesXup to isomorphism. The embedding{X\hookrightarrow G(X)}appears to be a universal (non-commutative) group-valued measure onX, while states ofXturn into real-valued group homomorphisms on{G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies toXand{G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of{G(X)}is shown to be the initial object among generalized Baer{{}^{\ast}}-semigroups. For finiteX, the correspondence betweenXand{G(X)}provides a new class of Garside groups.

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29

Gebhardt, Volker. "A new approach to the conjugacy problem in Garside groups." Journal of Algebra 292, no. 1 (October 2005): 282–302. http://dx.doi.org/10.1016/j.jalgebra.2005.02.002.

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30

Rump, Wolfgang. "Right l-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation." Journal of Algebra 439 (October 2015): 470–510. http://dx.doi.org/10.1016/j.jalgebra.2015.04.045.

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31

Lee, Eon-Kyung, and Sang-Jin Lee. "A Garside-theoretic approach to the reducibility problem in braid groups." Journal of Algebra 320, no. 2 (July 2008): 783–820. http://dx.doi.org/10.1016/j.jalgebra.2008.03.033.

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32

Crisp, John, and Luis Paris. "Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups." Pacific Journal of Mathematics 221, no. 1 (September 1, 2005): 1–27. http://dx.doi.org/10.2140/pjm.2005.221.1.

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33

Neaime, Georges. "Interval Garside structures for the complex braid groups $B(e,e,n)$." Transactions of the American Mathematical Society 372, no. 12 (September 6, 2019): 8815–48. http://dx.doi.org/10.1090/tran/7885.

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34

Digne, F. "A Garside presentation for Artin-Tits groups of type \widetilde{C}_n." Annales de l’institut Fourier 62, no. 2 (2012): 641–66. http://dx.doi.org/10.5802/aif.2690.

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35

Dehornoy, Patrick, Matthew Dyer, and Christophe Hohlweg. "Garside families in Artin–Tits monoids and low elements in Coxeter groups." Comptes Rendus Mathematique 353, no. 5 (May 2015): 403–8. http://dx.doi.org/10.1016/j.crma.2015.01.008.

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36

Licata, Anthony M., and Hoel Queffélec. "Braid groups of type ADE, Garside monoids, and the categorified root lattice." Annales Scientifiques de l'École Normale Supérieure 54, no. 2 (2021): 503–48. http://dx.doi.org/10.24033/asens.2464.

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37

DELOUP, FLORIAN. "PALINDROMES AND ORDERINGS IN ARTIN GROUPS." Journal of Knot Theory and Its Ramifications 19, no. 02 (February 2010): 145–62. http://dx.doi.org/10.1142/s0218216510007802.

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The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

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38

Corran, Ruth, and Matthieu Picantin. "A new Garside structure for the braid groups of type (e, e, r )." Journal of the London Mathematical Society 84, no. 3 (September 20, 2011): 689–711. http://dx.doi.org/10.1112/jlms/jdr030.

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39

Dietzel, Carsten, Wolfgang Rump, and Xia Zhang. "One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups." Journal of Algebra 526 (May 2019): 51–80. http://dx.doi.org/10.1016/j.jalgebra.2019.02.012.

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40

Kim, Djun Maximilian, and Dale Rolfsen. "An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements." Canadian Journal of Mathematics 55, no. 4 (August 1, 2003): 822–38. http://dx.doi.org/10.4153/cjm-2003-034-2.

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AbstractWe define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

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41

Birman, Joan S., and Nancy C. Wrinkle. "Holonomic and Legendrian parametrizations of knots." Journal of Knot Theory and Its Ramifications 09, no. 03 (May 2000): 293–309. http://dx.doi.org/10.1142/s0218216500000141.

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Holonomic parametrizations of knots were introduced in 1997 by Vassiliev, who proved that every knot type can be given a holonomic parametrization. Our main result is that any two holonomic knots which represent the same knot type are isotopic in the space of holonomic knots. A second result emerges through the techniques used to prove the main result: strong and unexpected connections between the topology of knots and the algebraic solution to the conjugacy problem in the braid groups, via the work of Garside. We also discuss related parametrizations of Legendrian knots, and uncover connections between the concepts of holonomic and Legendrian parametrizations of knots.

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42

Kwun, Young, Abdul Nizami, Mobeen Munir, Zaffar Iqbal, Dishya Arshad, and Shin Min Kang. "Khovanov Homology of Three-Strand Braid Links." Symmetry 10, no. 12 (December 5, 2018): 720. http://dx.doi.org/10.3390/sym10120720.

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Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.

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43

DEHORNOY, P. "Groupes de Garside." Annales Scientifiques de l’École Normale Supérieure 35, no. 2 (2002): 267–306. http://dx.doi.org/10.1016/s0012-9593(02)01090-x.

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44

Boyer, P., M. Dàvila, C. Schaub, and J. Nassiet. "Growth hormone response to clonidine stimulation in depressive states — First part of a two-part study —." Psychiatry and Psychobiology 1, no. 3 (1986): 189–95. http://dx.doi.org/10.1017/s0767399x00000031.

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Summary31 patients presenting a Major Depressive Episode were divided into two groups (endogenous versus neurotic depression), in keeping with the Newcastle criteria (Garside and Roth, 1974). 10 patients were allocated to each group for the realization of a test of growth hormone (GH) response to clonidine stimulation.The patients received no psychotropes for 8 days prior to the test. Mean age for the neurotic and endogenous groups was respectively 36.67 ± 3.18 years and 44.71 ± 2.56 years. Severity of depression, assessed with the Hamilton rating scale (21 items), was comparable in the two groups (35.35 ± 4.12 versus 39.8 ± 6.13).The test was carried out in two phases in each patient. During the initial phase, saline was infused over 10 minutes and continuous sampling was realized over a 4-hour period (automatic fraction collector with peristaltic pump), at between 9 AM and 1 PM. 48 hours later, the same procedure was repeated with the addition of 15 µg clonidine. This procedure allowed partial neutralization, for interpretation of results, of the «test apprehension» effect. Assays were carried out by radioimmunoassay (pooled samples), and correspond to real values for 10-minute time intervals (integrated values).Spontaneous GH secretion in the endogenous group was significantly lower (0.57 ± 0.16 ng/ml) than in the neurotic group (5.03 ± 1.08 ng/ml) and the control group (2.47 ± 0.78 ng/ml). After clonidine stimulation, GH response in the neurotic group was identical to that in the control group. No significant response was observed in the endogenous group. These results confirm those of several previous studies (Matussek, Charney, Checkley, Boyer, Corn, Siever) and seem to indicate hyposensitivity of the post-synaptic α2-adrenergic receptors in endogenous depression. Nevertheless, spontaneous GH hyposecretion in the same patients necessarily involves other mechanisms. The hypotheses concerning these mechanisms will be discussed in the second part of this study.

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45

Kalka, Arkadius, Eran Liberman, and Mina Teicher. "Subgroup conjugacy problem for Garside subgroups of Garside groups." Groups – Complexity – Cryptology 2, no. 2 (January 2010). http://dx.doi.org/10.1515/gcc.2010.010.

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46

Chen, Xing, and Xuezhi Zhao. "Normal forms in braid groups with respect to some Gröbner–Shirshov basis." Journal of Knot Theory and Its Ramifications, December 11, 2020, 2043005. http://dx.doi.org/10.1142/s0218216520430051.

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In 2008, Bokut obtained a Gröbner–Shirshov basis [Formula: see text] of the braid group [Formula: see text] in the Artin–Garside generators and showed that [Formula: see text]-irreducible words of the [Formula: see text] coincided with the Garside normal forms of words. Using this basis [Formula: see text], we obtain the concrete expression of the [Formula: see text]-irreducible words, i.e. normal forms, of the [Formula: see text], and hence give a new understanding of the word problem and Garside normal forms of braid groups.

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47

Franco, Nuno, and Juan González-Meneses. "Computation of Centralizers in Braid groups and Garside groups." Revista Matemática Iberoamericana, 2003, 367–84. http://dx.doi.org/10.4171/rmi/352.

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48

Holt, Derek. "Garside groups have the falsification by fellow-traveller property." Groups, Geometry, and Dynamics, 2010, 777–84. http://dx.doi.org/10.4171/ggd/105.

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49

Birman, Joan, Volker Gebhardt, and Juan González-Meneses. "Conjugacy in Garside groups I: cyclings, powers and rigidity." Groups, Geometry, and Dynamics, 2007, 221–79. http://dx.doi.org/10.4171/ggd/12.

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50

Picantin, Matthieu. "Cyclic amalgams, HNN extensions, and Garside one-relator groups." Journal of Algebra, April 2021. http://dx.doi.org/10.1016/j.jalgebra.2021.03.022.

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