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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.

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1.Arzhantseva, G. and Drutu, C., Geometry of infinitely presented small cancellation groups, rapid decay and quasi-homomorphisms, arXiv:1212.5280.
2.Bavard, C., Longueur stable des commutateurs, Enseign. Math. (2) 37 (1–2) (1991), 109150.
3.Behrstock, J. and Charney, R., Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2) (2012), 339356.
4.Bestvina, M., Bromberg, K. and Fujiwara, K., Stable commutator length on mapping class groups, arXiv:1306.2394.
5.Bestvina, M. and Fujiwara, K., Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 6989 (electronic).
6.Bestvina, M. and Fujiwara, K., A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (1) (2009), 1140.
7.Birman, J., Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213238.
8.Bou-Rabee, K. and Hadari, A., Simple closed curves, word length and nilpotent quotients of free groups, Pacific J. Math. 254 (1) (2011), 6772.
9.Brandenbursky, M., Bi-invariant metrics and quasi-morphisms on groups of hamiltonian diffeomorphisms of surfaces, arXiv:1306.3350.
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), 795816.
11.Brooks, R., Some remarks on bounded cohomology, Ann. Math. Stud. 97 (1981), 5363.
12.Burago, D., Ivanov, S. and Polterovich, L., Conjugation-invariant norms on groups of geometric origin, Groups Diffeomorphisms 52 (2008), 221250.
13.Calegari, D., Word length in surface groups with characteristic generating sets, Proc. Amer. Math. Soc. 136 (7) (2008), 26312637.
14.Calegari, D., scl, MSJ Memoirs, vol. 20 (Mathematical Society of Japan, Tokyo, 2009).
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), 145169.
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), 12961319.
17.Davis, M. W., The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32 (Princeton University Press, Princeton, NJ, 2008).
18.Dyer, M. J., On minimal lengths of expressions of Coxeter group elements as products of reflections, Proc. Amer. Math. Soc. 129 (9) (2001), 25912595 (electronic).
19.Entov, M. and Polterovich, L., Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003), 16351676.
20.Epstein, D. and Fujiwara, K., The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), 12751289.
21.Gal, Ś. R. and Kędra, J., On bi-invariant word metrics, J. Topol. Anal. 3 (2) (2011), 161175.
22.Gambaudo, J.-M. and Ghys, E., Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Syst. 24 (5) (2004), 15911617.
23.Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1–2) (1990), 2538.
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), 213232 (2012).
25.Kotschick, D., Stable length in stable groups, Groups Diffeomorphisms 52 (2008), 4014113.
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).
27.Lalonde, F. and McDuff, D., The geometry of symplectic energy, Ann. Math. 141 (2) (1995), 349371.
28.Liehl, B., Beschränkte Wortlänge in SL2, Math. Z. 186 (4) (1984), 509524.
29.Magnus, W., Über automorphismen von fundamentalgruppen berandeter flächen, Math. Ann. 109 (1934), 617646.
30.Marcinkowski, M., Programm for computing the biinvariant norm. Available at:
31.McCammond, J. and Petersen, T. K., Bounding reflection length in an affine Coxeter group, J. Algebr. Combin. 34 (4) (2011), 711719.
32.Muranov, A., Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length, J. Topol. Anal. 2 (3) (2010), 341384.
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), 609619.
34.Rolfsen, D. and Zhu, J., Braids, orderings and zero divisors, J. Knot Theory Ramifications 7 (6) (1998), 837841.
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).
36.Sury, B., Bounded generation does not imply finite presentation, Comm. Algebra 25 (5) (1997), 16731683.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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