Geometry of Infinitely Presented Small Cancellation Groups and Quasi-homomorphisms

Arzhantseva, Goulnara ; Druţu, Cornelia

Canadian journal of mathematics, Tome 71 (2019) no. 5, pp. 997-1018

Résumé

We study the geometry of infinitely presented groups satisfying the small cancellation condition $C^{\prime }(1/8)$, and introduce a standard decomposition (called the criss-cross decomposition) for the elements of such groups. Our method yields a direct construction of a linearly independent set of power continuum in the kernel of the comparison map between the bounded and the usual group cohomology in degree 2, without the use of free subgroups and extensions.

Détail

DOI : 10.4153/CJM-2018-036-7

Mots-clés : small cancellation theory, Greendlinger lemma, quasi-homomorphism, bounded cohomology

Bibliographie

[Bav91] Bavard, C., Longueur stable des commutateurs . Enseign. Math. (2) 37(1991), no. 1–2, 109–150. Google Scholar

[BF02] Bestvina, M. and Fujiwara, K., Bounded cohomology of subgroups of mapping class groups . Geom. Topol. 6(2002), 69–89 (electronic). . Google Scholar | DOI

[BM99] Burger, M. and Monod, N., Bounded cohomology of lattices in higher rank Lie groups . J. Eur. Math. Soc. (JEMS) 1(1999), no. 2, 199–235. . Google Scholar | DOI

[BM02] Burger, M. and Monod, N., Continuous bounded cohomology and applications to rigidity theory . Geom. Funct. Anal. 12(2002), no. 2, 219–280. . Google Scholar | DOI

[Bro81] Brooks, R., Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 53–63. Google Scholar

[DGO17] Dahmani, F., Guirardel, V., and Osin, D., Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces . Mem. Amer. Math. Soc. 245(2017), no. 1156. . Google Scholar | DOI

[EF97] Epstein, D. B. A. and Fujiwara, K., The second bounded cohomology of word-hyperbolic groups . Topology 36(1997), no. 6, 1275–1289. . Google Scholar | DOI

[Fuj98] Fujiwara, K., The second bounded cohomology of a group acting on a Gromov-hyperbolic space . Proc. London Math. Soc. (3) 76(1998), no. 1, 70–94. . Google Scholar | DOI

[Fuj00] Fujiwara, K., The second bounded cohomology of an amalgamated free product of groups . Trans. Amer. Math. Soc. 352(2000), no. 3, 1113–1129. . Google Scholar | DOI

[Fuj09] Fujiwara, K., Quasi-homomorphisms on mapping class groups. In: Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009, pp. 241–269. Google Scholar

[GdlH91] Ghys, E. and De La Harpe, P., Infinite groups as geometric objects (after Gromov). In: Ergodic theory, symbolic dynamics and hyperbolic spaces (Trieste, 1989) Oxford University Press, 1991, pp. 299–314. Google Scholar

[Gro82] Gromov, M., Volume and bounded cohomology . Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99. Google Scholar

[Gro87] Gromov, M., Hyperbolic groups. In: Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. . Google Scholar | DOI

[GS14] Gruber, D. and Sisto, A., Infinitely presented graphical small cancellation groups are acylindrically hyperbolic. Ann. Inst. Fourier., to appear. arxiv:1408.4488. Google Scholar

[HO13] Hull, M. and Osin, D., Induced quasicocycles on groups with hyperbolically embedded subgroups . Algebr. Geom. Topol. 13(2013), no. 5, 2635–2665. . Google Scholar | DOI

[LS77] Lyndon, R. and Schupp, P., Combinatorial group theory . Springer-Verlag, Berlin-New York, 1977. Google Scholar

[RS95] Rips, E. and Sela, Z., Canonical representatives and equations in hyperbolic groups . Invent. Math. 120(1995), no. 3, 489–512. . Google Scholar | DOI

[Sel92] Sela, Z., Uniform embeddings of hyperbolic groups in Hilbert spaces . Israel J. Math. 80(1992), no. 1–2, 171–181. . Google Scholar | DOI