Groups with graphical 𝐶(6) and 𝐶(7) small cancellation presentations

Gruber, Dominik

doi:10.1090/S0002-9947-2014-06198-9

Groups with graphical $C(6)$ and $C(7)$ small cancellation presentations
HTML articles powered by AMS MathViewer

by Dominik Gruber PDF

Trans. Amer. Math. Soc. 367 (2015), 2051-2078 Request permission

Abstract:

We extend fundamental results of small cancellation theory to groups whose presentations satisfy the generalizations of the classical $C(6)$ and $C(7)$ conditions in graphical small cancellation theory. Using these graphical small cancellation conditions, we construct lacunary hyperbolic groups and groups that coarsely contain prescribed infinite sequences of finite graphs. We prove that groups given by (possibly infinite) graphical $C(7)$ presentations contain non-abelian free subgroups.

References

J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on word hyperbolic groups, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 3–63. Edited by Short. MR 1170363
G. N. Arzhantseva, T. Delzant, Examples of random groups, preprint (2008).
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Ian M. Chiswell, Donald J. Collins, and Johannes Huebschmann, Aspherical group presentations, Math. Z. 178 (1981), no. 1, 1–36. MR 627092, DOI 10.1007/BF01218369
D. J. Collins and J. Huebschmann, Spherical diagrams and identities among relations, Math. Ann. 261 (1982), no. 2, 155–183. MR 675732, DOI 10.1007/BF01456216
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146. MR 1978492, DOI 10.1007/s000390300002
N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354. MR 1911663, DOI 10.1007/s00039-002-8249-5
Ilya Kapovich and Alexei Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002), no. 2, 608–668. MR 1882114, DOI 10.1006/jabr.2001.9033
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Jonathan P. McCammond and Daniel T. Wise, Fans and ladders in small cancellation theory, Proc. London Math. Soc. (3) 84 (2002), no. 3, 599–644. MR 1888425, DOI 10.1112/S0024611502013424
Yann Ollivier, On a small cancellation theorem of Gromov, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), no. 1, 75–89. MR 2245980
Yann Ollivier and Daniel T. Wise, Kazhdan groups with infinite outer automorphism group, Trans. Amer. Math. Soc. 359 (2007), no. 5, 1959–1976. MR 2276608, DOI 10.1090/S0002-9947-06-03941-9
A. Yu. Ol′shanskiĭ, Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin. MR 1191619, DOI 10.1007/978-94-011-3618-1
Alexander Yu. Ol′shanskii, Denis V. Osin, and Mark V. Sapir, Lacunary hyperbolic groups, Geom. Topol. 13 (2009), no. 4, 2051–2140. With an appendix by Michael Kapovich and Bruce Kleiner. MR 2507115, DOI 10.2140/gt.2009.13.2051
Eliyahu Rips and Yoav Segev, Torsion-free group without unique product property, J. Algebra 108 (1987), no. 1, 116–126. MR 887195, DOI 10.1016/0021-8693(87)90125-6
L. Silberman, Addendum to: “Random walk in random groups” [Geom. Funct. Anal. 13 (2003), no. 1, 73–146; MR1978492] by M. Gromov, Geom. Funct. Anal. 13 (2003), no. 1, 147–177. MR 1978493, DOI 10.1007/s000390300003
Ralph Strebel, Appendix. Small cancellation groups, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) Progr. Math., vol. 83, Birkhäuser Boston, Boston, MA, 1990, pp. 227–273. MR 1086661