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THE CANCELLATION NORM AND THE GEOMETRY OF BI-INVARIANT WORD METRICS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  21 July 2015

MICHAEL BRANDENBURSKY ,
ŚWIATOSŁAW R. GAL ,
JAREK KĘDRA  and
MICHAŁ MARCINKOWSKI
MICHAEL BRANDENBURSKY
Affiliation:
CRM, University of Montreal, Canada e-mail: [email protected]
ŚWIATOSŁAW R. GAL
Affiliation:
Instytut Matematyczny Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland e-mail: [email protected]
JAREK KĘDRA
Affiliation:
Institute of Pure and Applied Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, Scotland e-mail: [email protected]
MICHAŁ MARCINKOWSKI
Affiliation:
Instytut Matematyczny Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland e-mail: [email protected]
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Abstract

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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.

MSC classification

Primary: 20F65: Geometric group theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 1 , January 2016 , pp. 153 - 176
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Arzhantseva, G. and Drutu, C., Geometry of infinitely presented small cancellation groups, rapid decay and quasi-homomorphisms, arXiv:1212.5280.Google Scholar
2.Bavard, C., Longueur stable des commutateurs, Enseign. Math. (2) 37 (1–2) (1991), 109–150.Google Scholar
3.Behrstock, J. and Charney, R., Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2) (2012), 339–356.CrossRefGoogle Scholar
4.Bestvina, M., Bromberg, K. and Fujiwara, K., Stable commutator length on mapping class groups, arXiv:1306.2394.Google Scholar
5.Bestvina, M. and Fujiwara, K., Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89 (electronic).CrossRefGoogle Scholar
6.Bestvina, M. and Fujiwara, K., A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (1) (2009), 11–40.CrossRefGoogle Scholar
7.Birman, J., Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
9.Brandenbursky, M., Bi-invariant metrics and quasi-morphisms on groups of hamiltonian diffeomorphisms of surfaces, arXiv:1306.3350.Google Scholar
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.CrossRefGoogle Scholar
11.Brooks, R., Some remarks on bounded cohomology, Ann. Math. Stud. 97 (1981), 53–63.Google Scholar
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.Calegari, D., Word length in surface groups with characteristic generating sets, Proc. Amer. Math. Soc. 136 (7) (2008), 2631–2637.CrossRefGoogle Scholar
14.Calegari, D., scl, MSJ Memoirs, vol. 20 (Mathematical Society of Japan, Tokyo, 2009).CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.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).CrossRefGoogle Scholar
19.Entov, M. and Polterovich, L., Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003), 1635–1676.CrossRefGoogle Scholar
20.Epstein, D. and Fujiwara, K., The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), 1275–1289.CrossRefGoogle Scholar
21.Gal, Ś. R. and Kędra, J., On bi-invariant word metrics, J. Topol. Anal. 3 (2) (2011), 161–175.CrossRefGoogle Scholar
22.Gambaudo, J.-M. and Ghys, E., Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Syst. 24 (5) (2004), 1591–1617.CrossRefGoogle Scholar
23.Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1–2) (1990), 25–38.CrossRefGoogle Scholar
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.Kotschick, D., Stable length in stable groups, Groups Diffeomorphisms 52 (2008), 401–4113.Google Scholar
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.Lalonde, F. and McDuff, D., The geometry of symplectic energy, Ann. Math. 141 (2) (1995), 349–371.CrossRefGoogle Scholar
28.Liehl, B., Beschränkte Wortlänge in SL2, Math. Z. 186 (4) (1984), 509–524.CrossRefGoogle Scholar
29.Magnus, W., Über automorphismen von fundamentalgruppen berandeter flächen, Math. Ann. 109 (1934), 617–646.CrossRefGoogle Scholar
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.McCammond, J. and Petersen, T. K., Bounding reflection length in an affine Coxeter group, J. Algebr. Combin. 34 (4) (2011), 711–719.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
34.Rolfsen, D. and Zhu, J., Braids, orderings and zero divisors, J. Knot Theory Ramifications 7 (6) (1998), 837–841.CrossRefGoogle Scholar
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.Sury, B., Bounded generation does not imply finite presentation, Comm. Algebra 25 (5) (1997), 1673–1683.CrossRefGoogle Scholar