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)

 
 

 

A group theoretic criterion for property FA


Authors: Marc Culler and Karen Vogtmann
Journal: Proc. Amer. Math. Soc. 124 (1996), 677-683
MSC (1991): Primary 20E08; Secondary 20F28, 20F36, 05C05
DOI: https://doi.org/10.1090/S0002-9939-96-03217-0
MathSciNet review: 1307506
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give group-theoretic conditions on a set of generators of a group $G$ which imply that $G$ admits no non-trivial action on a tree. The criterion applies to several interesting classes of groups, including automorphism groups of most free groups and mapping class groups of most surfaces.


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

  • [1] R. Alperin and H. Bass, Length functions of group actions on $\Lambda $-trees, Combinatorial Group Theory and Topology (S. Gersten and J. Stallings, eds.), Ann. of Math. Stud., vol. III, Princeton Univ. Press, Princeton, NJ, 1986, pp. 265--378. MR 89c:20057
  • [2] H. Bass, Algebraic K-theory, Benjamin, New York, 1968. MR 40:2736
  • [3] R. Bieri, W. D. Neumann, and R. Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), 451--477. MR 89b:20108
  • [4] O. V. Bogopolski, Arboreal decomposability of the group of automorphisms of a free group, Algebra i Logica 26 (1987), no. 2, 131--149.
  • [5] K. Brown, Trees, valuations and the Bieri-Neumann-Strebel invariant, Invent. Math. 90 (1987), 479--504. MR 89e:20060
  • [6] M. Culler and J. Morgan, Group actions on ${\mathbb R} $-trees, Proc. London Math. Soc. (3) 55 (1987), 571--604. MR 88f:20055
  • [7] N. V. Ivanov, Complexes of curves and the Teichmüller modular group, Russian Math. Surveys 42 (1987), no 3, 55--107. MR 88g:32040
  • [8] D. Mumford, Abelian quotients of the Teichmüller modular group, J. Analyse Math. 18 (1967), 227--224. MR 36:2623
  • [9] J.-P. Serre, Arbres, amalgames, $SL_2$, Asterisque 46 (1977). MR 57:16426
  • [10] J. Tits, A theorem of Lie-Kolchin for trees, Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin (H. Bass, P. J. Cassidy, and J. Kovacic, eds.), Academic Press, New York, 1977, pp. 377--388. MR 58:28205

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20E08, 20F28, 20F36, 05C05

Retrieve articles in all journals with MSC (1991): 20E08, 20F28, 20F36, 05C05


Additional Information

Marc Culler
Affiliation: Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607-7045
Email: culler@math.uic.edu

Karen Vogtmann
Affiliation: Department of Mathematics, Cornell University, White Hall, Ithaca, New York 14853-7901
Email: vogtmann@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03217-0
Keywords: Property FA, group actions on trees
Received by editor(s): May 31, 1994
Additional Notes: Both authors are partially supported by the National Science Foundation.
Communicated by: James West
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society