Small cancellation groups and translation numbers

Kapovich, Ilya

doi:10.1090/S0002-9947-97-01653-X

Small cancellation groups and translation numbers
HTML articles powered by AMS MathViewer

by Ilya Kapovich PDF

Trans. Amer. Math. Soc. 349 (1997), 1851-1875 Request permission

Abstract:

In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any $g$ and any $n$ there is an algorithm deciding whether or not the equation $x^{n}=g$ has a solution. There is also an algorithm for calculating for each $g$ the maximum $n$ such that $g$ is an $n$-th power of some element. We also note that these groups cannot contain isomorphic copies of the group of $p$-adic fractions and so in particular of the group of rational numbers. Besides we show that for $C^{\prime \prime }(4)-T(4)$ and $C''(3)-T(6)$ groups all translation numbers are rational and have bounded denominators.

References

Michael Anshel and Peter Stebe, Conjugate powers in free products with amalgamation, Houston J. Math. 2 (1976), no. 2, 139–147. MR 409664
K. I. Appel and P. E. Schupp, Artin groups and infinite Coxeter groups, Invent. Math. 72 (1983), no. 2, 201–220. MR 700768, DOI 10.1007/BF01389320
W.Ballmann, Singular spaces of non-positive curvature, Sur les Groupes Hyperboliques d’apres Mikhael Gromov (E.Ghys and P. de la Harpe, eds.), Birkhauser, Berlin, 1990 pages 189–201.
W. Ballmann and M. Brin, Polygonal complexes and combinatorial group theory, Geom. Dedicata 50 (1994), no. 2, 165–191. MR 1279883, DOI 10.1007/BF01265309
Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991), no. 3, 229–316. MR 1147304, DOI 10.1016/0022-4049(91)90139-S
Martin R. Bridson, Geodesics and curvature in metric simplicial complexes, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 373–463. MR 1170372
M.R.Bridson and A.Haefliger, An introduction to CAT(0)-spaces, in preparation.
Leo P. Comerford Jr., Powers and conjugacy in small cancellation groups, Arch. Math. (Basel) 26 (1975), no. 4, 353–360. MR 374279, DOI 10.1007/BF01229751
Leo P. Comerford Jr., A note on power-conjugacy, Houston J. Math. 3 (1977), no. 3, 337–341. MR 450406
G.Conner, Metrics on Groups, PhD Thesis, University of Utah, 1992.
G.Conner, Properties of translation numbers in solvable groups, Brigham Young University, preprint (1994).
G.Conner, A finitely generated group with irrational translation numbers, Brigham Young University, preprint (1994).
Leo P. Comerford Jr. and Bernard Truffault, The conjugacy problem for free products of sixth-groups with cyclic amalgamation, Math. Z. 149 (1976), no. 2, 169–181. MR 409666, DOI 10.1007/BF01301574
David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694, DOI 10.1201/9781439865699
S. M. Gersten and H. B. Short, Small cancellation theory and automatic groups, Invent. Math. 102 (1990), no. 2, 305–334. MR 1074477, DOI 10.1007/BF01233430
S. M. Gersten and H. B. Short, Rational subgroups of biautomatic groups, Ann. of Math. (2) 134 (1991), no. 1, 125–158. MR 1114609, DOI 10.2307/2944334
S. M. Gersten and H. Short, Small cancellation theory and automatic groups. II, Invent. Math. 105 (1991), no. 3, 641–662. MR 1117155, DOI 10.1007/BF01232283
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}
K. Johnsgard, Geodesic tilings for equal geodesic words in $C^{\prime \prime }(p)-T(q)$ group presentations, Cornell University, preprint, 1993.
K. Johnsgard, Two automatic spanning trees in small cancellation group presentations, Intern. J. Alg. Comp. (to appear).
Seymour Lipschutz, On Greendlinger groups, Comm. Pure Appl. Math. 23 (1970), 743–747. MR 271203, DOI 10.1002/cpa.3160230503
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
Mohamed S. El-Mosalamy and Stephen J. Pride, On $T(6)$-groups, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 3, 443–451. MR 906618, DOI 10.1017/S0305004100067499
E.Swenson, Hyperbolic Elements in Negatively Curved Groups, Geom. Dedicata 55 (1995), 199–210.
C. M. Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. Soc. 30 (1971), 22–26. MR 279169, DOI 10.1090/S0002-9939-1971-0279169-X