Geodesic
THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS
Glasgow mathematical journal, Tome 58 (2016) no. 1, pp. 153-176

Voir la notice de l'article provenant de la source Cambridge University Press

We study bi-invariant word metrics on groups. We provide an efficient algorithm for computing the bi-invariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the bi-invariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups, cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin. 
BRANDENBURSKY, MICHAEL; GAL, ŚWIATOSŁAW R.; KĘDRA, JAREK; MARCINKOWSKI, MICHAŁ. THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS. Glasgow mathematical journal, Tome 58 (2016) no. 1, pp. 153-176. doi: 10.1017/S0017089515000129
@article{10_1017_S0017089515000129,
     author = {BRANDENBURSKY, MICHAEL and GAL, \'SWIATOS{\L}AW R. and K\k{E}DRA, JAREK and MARCINKOWSKI, MICHA{\L}},
     title = {THE {CANCELLATION} {NORM} {AND} {THE} {GEOMETRY} {OF} {BI-INVARIANT} {WORD} {METRICS}},
     journal = {Glasgow mathematical journal},
     pages = {153--176},
     year = {2016},
     volume = {58},
     number = {1},
     doi = {10.1017/S0017089515000129},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000129/}
}
TY  - JOUR
AU  - BRANDENBURSKY, MICHAEL
AU  - GAL, ŚWIATOSŁAW R.
AU  - KĘDRA, JAREK
AU  - MARCINKOWSKI, MICHAŁ
TI  - THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS
JO  - Glasgow mathematical journal
PY  - 2016
SP  - 153
EP  - 176
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000129/
DO  - 10.1017/S0017089515000129
ID  - 10_1017_S0017089515000129
ER  - 
%0 Journal Article
%A BRANDENBURSKY, MICHAEL
%A GAL, ŚWIATOSŁAW R.
%A KĘDRA, JAREK
%A MARCINKOWSKI, MICHAŁ
%T THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS
%J Glasgow mathematical journal
%D 2016
%P 153-176
%V 58
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000129/
%R 10.1017/S0017089515000129
%F 10_1017_S0017089515000129

[1] 1.Arzhantseva, G. and Drutu, C., Geometry of infinitely presented small cancellation groups, rapid decay and quasi-homomorphisms, arXiv:1212.5280. Google Scholar

[2] 2.Bavard, C., Longueur stable des commutateurs, Enseign. Math. (2) 37 (1–2) (1991), 109–150. Google Scholar

[3] 3.Behrstock, J. and Charney, R., Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2) (2012), 339–356. Google Scholar | DOI

[4] 4.Bestvina, M., Bromberg, K. and Fujiwara, K., Stable commutator length on mapping class groups, arXiv:1306.2394. Google Scholar

[5] 5.Bestvina, M. and Fujiwara, K., Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89 (electronic). Google Scholar | DOI

[6] 6.Bestvina, M. and Fujiwara, K., A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (1) (2009), 11–40. Google Scholar | DOI

[7] 7.Birman, J., Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238. Google Scholar | DOI

[8] 8.Bou-Rabee, K. and Hadari, A., Simple closed curves, word length and nilpotent quotients of free groups, Pacific J. Math. 254 (1) (2011), 67–72. Google Scholar | DOI

[9] 9.Brandenbursky, M., Bi-invariant metrics and quasi-morphisms on groups of hamiltonian diffeomorphisms of surfaces, arXiv:1306.3350. Google Scholar

[10] 10.Brandenbursky, M. and Kędra, J., On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc, Algebr. Geom. Topology 13 (2013), 795–816. Google Scholar | DOI

[11] 11.Brooks, R., Some remarks on bounded cohomology, Ann. Math. Stud. 97 (1981), 53–63. Google Scholar

[12] 12.Burago, D., Ivanov, S. and Polterovich, L., Conjugation-invariant norms on groups of geometric origin, Groups Diffeomorphisms 52 (2008), 221–250. Google Scholar

[13] 13.Calegari, D., Word length in surface groups with characteristic generating sets, Proc. Amer. Math. Soc. 136 (7) (2008), 2631–2637. Google Scholar | DOI

[14] 14.Calegari, D., scl, MSJ Memoirs, vol. 20 (Mathematical Society of Japan, Tokyo, 2009). Google Scholar | DOI

[15] 15.Calegari, D. and Zhuang, D., Stable W-length, in Topology and geometry in dimension three, Contemp. Math., vol. 560 American Mathematical Society, Providence, RI, 2011), 145–169. Google Scholar | DOI

[16] 16.Caprace, P.-E. and Fujiwara, K., Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (5) (2010), 1296–1319. Google Scholar | DOI

[17] 17.Davis, M. W., The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32 (Princeton University Press, Princeton, NJ, 2008). Google Scholar

[18] 18.Dyer, M. J., On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (9) (2001), 2591–2595 (electronic). Google Scholar | DOI

[19] 19.Entov, M. and Polterovich, L., Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003), 1635–1676. Google Scholar | DOI

[20] 20.Epstein, D. and Fujiwara, K., The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), 1275–1289. Google Scholar | DOI

[21] 21.Gal, Ś. R. and Kędra, J., On bi-invariant word metrics, J. Topol. Anal. 3 (2) (2011), 161–175. Google Scholar | DOI

[22] 22.Gambaudo, J.-M. and Ghys, E., Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Syst. 24 (5) (2004), 1591–1617. Google Scholar | DOI

[23] 23.Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1–2) (1990), 25–38. Google Scholar | DOI

[24] 24.Kaabi, N. and Vershinin, V., On Vassiliev invariants of braid groups of the sphere, (English summary) Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 58 (2011), 213–232 (2012). Google Scholar

[25] 25.Kotschick, D., Stable length in stable groups, Groups Diffeomorphisms 52 (2008), 401–4113. Google Scholar

[26] 26.Kurosh, A. G., The theory of groups. vol. II, Translated from the Russian and edited by Hirsch, K. A. (Chelsea Publishing Company, New York, N.Y., 1956). Google Scholar

[27] 27.Lalonde, F. and Mcduff, D., The geometry of symplectic energy, Ann. Math. 141 (2) (1995), 349–371. Google Scholar | DOI

[28] 28.Liehl, B., Beschränkte Wortlänge in SL, Math. Z. 186 (4) (1984), 509–524. Google Scholar | DOI

[29] 29.Magnus, W., Über automorphismen von fundamentalgruppen berandeter flächen, Math. Ann. 109 (1934), 617–646. Google Scholar | DOI

[30] 30.Marcinkowski, M., Programm for computing the biinvariant norm. Available at: http://www.math.uni.wroc.pl/~marcinkow/papers/program.biinv/biinv.length.v1.0.tar. Google Scholar

[31] 31.Mccammond, J. and Petersen, T. K., Bounding reflection length in an affine Coxeter group, J. Algebr. Combin. 34 (4) (2011), 711–719. Google Scholar | DOI

[32] 32.Muranov, A., Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length, J. Topol. Anal. 2 (3) (2010), 341–384. Google Scholar | DOI

[33] 33.Polterovich, L. and Rudnick, Z., Stable mixing for cat maps and quasi-morphisms of the modular group, Ergodic Theory Dynam. Syst. 24 (2) (2004), 609–619. Google Scholar | DOI

[34] 34.Rolfsen, D. and Zhu, J., Braids, orderings and zero divisors, J. Knot Theory Ramifications 7 (6) (1998), 837–841. Google Scholar | DOI

[35] 35.Serre, J.-P., Trees, Springer Monographs in Mathematics, Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation, (Springer-Verlag, Berlin, 2003). Google Scholar

[36] 36.Sury, B., Bounded generation does not imply finite presentation, Comm. Algebra 25 (5) (1997), 1673–1683. Google Scholar | DOI

Cité par Sources :