Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Representation of tree permutations by words


Author: John A. Maroli
Journal: Proc. Amer. Math. Soc. 110 (1990), 859-869
MSC: Primary 06A06; Secondary 06F15, 20B27
DOI: https://doi.org/10.1090/S0002-9939-1990-1037214-X
MathSciNet review: 1037214
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of solving equations in groups can be stated as follows: given a group $ G$ and a free group $ F = F\left( {{x_1},{x_2}, \ldots } \right)$, for which pairs $ \left( {w,g} \right)$ with $ w = w\left( {{x_1},{x_2}, \ldots } \right) \in F,g \in G$, is it possible to find elements $ {g_i} \in G$ such that $ w\left( {{g_1},{g_2}, \ldots } \right) = g$? We investigate the corresponding question of solving equations in the group $ A\left( \Omega \right)$ of all automorphisms of a transitive tree $ \Omega $. If the tree has isomorphic cones at a branch point, then certain equations of the form $ {x^n} = g$ cannot be solved (Theorem 2.3). If the tree is sufficiently transitive, we find large classes of equations $ w = g$ which can be solved (Theorems 2.13, 2.16).


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

  • [1] S. A. Adeleke and W. C. Holland, Representation of order automorphisms by words, preprint, 1987. MR 1269842 (95c:20042)
  • [2] M. Droste, Structure of partially ordered sets with transitive automorphism groups, Mem. Amer. Math. Soc., no. 334, Amer. Math. Soc., Providence, RI, 1985. MR 803976 (87d:06005)
  • [3] M. Droste, W. C. Holland, and H. D. Macpherson, Automorphism groups of infinite semilinear orders (I), Proc. London Math. Soc. 58 (1989), 454-478. MR 988099 (90b:20006)
  • [4] A. M. W. Glass, Ordered permutation groups, London Math. Soc. Lecture Note Ser. 55 (1981). MR 645351 (83j:06004)
  • [5] W. C. Holland, Transitive lattice-ordered permutation groups, Math. Z. 87 (1965), 420-433. MR 0178052 (31:2310)
  • [6] R. C. Lyndon, Words and infinite permutations, preprint, 1986. MR 1252660 (95c:20006)
  • [7] J. A. Maroli, Tree permutation groups, Doctoral Dissertation, Bowling Green State University, 1989.
  • [8] S. H. McCleary, $ o$-primitive ordered permutation groups, Pacific J. Math. 40 (1972), 349-372. MR 0313155 (47:1710)
  • [9] J. Mycielski, Representations of infinite permutations by words, Proc. Amer. Math. Soc. 100 (1987), 237-241. MR 884459 (88c:20044)
  • [10] D. M. Silberger, Are primitive words universal for infinite symmetric groups?, Trans. Amer. Math. Soc. 276 (1983), 841-852. MR 688980 (84c:20011)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 06A06, 06F15, 20B27

Retrieve articles in all journals with MSC: 06A06, 06F15, 20B27


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1037214-X
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society