Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The solution of length four equations over groups


Authors: Martin Edjvet and James Howie
Journal: Trans. Amer. Math. Soc. 326 (1991), 345-369
MSC: Primary 20E06; Secondary 20F05, 20F06
DOI: https://doi.org/10.1090/S0002-9947-1991-1002920-5
MathSciNet review: 1002920
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a group, $ F$ the free group generated by $ t$ and let $ r(t) \in G \ast F$. The equation $ r(t) = 1$ is said to have a solution over $ G$ if it has a solution in some group that contains $ G$. This is equivalent to saying that the natural map $ G \to \langle G \ast F\vert r(t)\rangle $ is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of $ t$ in $ r(t)$ is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of $ t$ in $ r(t)$ is equal to four.


References [Enhancements On Off] (What's this?)

  • [1] D. J. Collins and J. Huebschmann, Spherical diagrams and identities among relations, Math. Ann. 261 (1982), 155-183. MR 675732 (84b:20035)
  • [2] M. Edjvet, Solutions of certain sets of equations over groups, St. Andrews 1990, London Math. Soc. Lecture Notes Series, Cambridge Univ. Press (to appear). MR 1123977 (93f:20055)
  • [3] M. Gerstenhaber and O. S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531-1533. MR 0166296 (29:3573)
  • [4] J. Howie, The solution of length three equations over groups, Proc. Edinburgh Math. Soc. (2) 26 (1983), 89-96. MR 695646 (85b:20047)
  • [5] F. Levin, Solutions of equations over groups, Bull. Amer. Math. Soc. 68 (1962), 603-604. MR 0142643 (26:212)
  • [6] R. C. Lyndon, Equations in groups, Bol. Soc. Brasil. Mat. 11 (1980), 79-102. MR 607019 (82j:20070)
  • [7] -, Equations over cyclic groups, Preprint, Université Paris VII, 1981.
  • [8] -, Problems in combinatorial group theory, Combinatorial Group Theory and Topology (S. M. Gersten and J. R. Stallings, eds.), Ann. of Math. Studies, no. 111, Princeton Univ. Press, Princeton, N.J., 1987, pp. 3-33. MR 895607 (88f:20003)
  • [9] B. H. Neumann, Adjunction of elements to groups, J. London Math. Soc. 18 (1942), 4-11. MR 0008808 (5:58s)
  • [10] O. S. Rothaus, On the nontriviality of some group extensions given by generators and relations, Ann. of Math. (2) 106 (1977), 559-612. MR 0467760 (57:7612)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20E06, 20F05, 20F06

Retrieve articles in all journals with MSC: 20E06, 20F05, 20F06


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1002920-5
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society