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Abelian subgroups of Garside groups. (English) Zbl 1155.20037
Summary: We show that for every Abelian subgroup $$H$$ of a Garside group, some conjugate $$g^{-1}Hg$$ consists of ultra summit elements and the centralizer of $$H$$ is a finite index subgroup of the normalizer of $$H$$. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning Abelian subgroups of Garside groups.

##### MSC:
 20F36 Braid groups; Artin groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth 20E45 Conjugacy classes for groups
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##### References:
 [1] Alonso J. M., Proc. London Math. Soc. 70 pp 56– (1995) · Zbl 0823.20035 [2] Bestvina M., Geometry and Topology 3 pp 269– (1999) · Zbl 0998.20034 [3] Birman J. S., Duke Math. J. 50 pp 1107– (1983) · Zbl 0551.57004 [4] Birman J. S., Adv. Math. 139 (2) pp 322– (1998) · Zbl 0937.20016 [5] Birman J. S., Groups, Geometry and Dynamics 1 (3) pp 221– (2006) [6] Bridson M. R., Metric Spaces of Non-Positive Curvature (1999) · Zbl 0988.53001 [7] Charney R., Commentari Math. Helv. 78 pp 584– (2003) · Zbl 1066.20043 [8] Charney R., Geom. Dedicata 105 pp 171– (2004) · Zbl 1064.20044 [9] Cohen H. A., A Course in Computational Algebraic Number Theory (1993) · Zbl 0786.11071 [10] Dehornoy P., J. Algebra 210 pp 291– (1998) · Zbl 0959.20035 [11] Dehornoy P., Ann. Scient. Ec. Norm Sup. 35 pp 267– (2002) [12] Dehornoy P., Proc. London Math. Soc. 79 (3) pp 569– (1999) · Zbl 1030.20021 [13] Epstein D. B. A., Word Processing in Groups (Chapter 9) (1992) · Zbl 0764.20017 [14] Franco N., J. Algebra 266 (1) pp 112– (2003) · Zbl 1043.20019 [15] Garside F. A., Quart. J. Math. Oxford Ser. 20 pp 235– (1969) · Zbl 0194.03303 [16] Gebhardt V., J. Algebra 292 (1) pp 282– (2005) · Zbl 1105.20032 [17] Gersten S. M., Ann. of Math. 134 (1) pp 125– (1991) · Zbl 0744.20035 [18] Gonzálezez-Meneses J., Contemp. Mathematics 372 pp 35– (2005) [19] Lee E.-K., J. Pure Appl. Algebra 211 (3) pp 732– (2006) · Zbl 1150.20022 [20] Lee S. J., J. Algebra 309 pp 594– (2007) · Zbl 1155.20038 [21] Lee , S. J. , Lee , E. ( 2002 ). Potential weaknesses of the commutator key agreement protocol based on braid groups .EUROCRYPT 2002. Lecture Notes in Computer Science 2332. Springer , pp. 14 – 28 . · Zbl 1055.94019
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