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

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Abstract | References | Similar Articles | Additional Information

Abstract: We give group-theoretic conditions on a set of generators of a group which imply that 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.

**[1]**R. Alperin and H. Bass,*Length functions of group actions on -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 -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,*, 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**

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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