Virtually free groups

with finitely many outer automorphisms

Author:
Martin R. Pettet

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4565-4587

MSC (1991):
Primary 20F28; Secondary 20E36, 20E08

MathSciNet review:
1370649

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finitely generated virtually free group. From a presentation of as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived for the outer automorphism group of to be finite. Two versions of the characterization are given, both effectively verifiable from the graph of groups. The more purely group theoretical criterion is expressed in terms of the structure of the normalizers of the edge groups (Theorem 5.10); the other version involves certain finiteness conditions on the associated -tree (Theorem 5.16). Coupled with an earlier result, this completes a description of the finitely generated groups whose full automorphism groups are virtually free.

**1.**J. L. Alperin,*Groups with finitely many automorphisms*, Pacific J. Math.**12**(1962), 1–5. MR**0140592****2.**Daniel E. Cohen,*Combinatorial group theory: a topological approach*, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989. MR**1020297****3.**Warren Dicks and M. J. Dunwoody,*Groups acting on graphs*, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR**1001965****4.**Sašo Kalajdžievski,*Automorphism group of a free group: centralizers and stabilizers*, J. Algebra**150**(1992), no. 2, 435–502. MR**1176906**, 10.1016/S0021-8693(05)80041-9**5.**A. Karrass, A. Pietrowski, and D. Solitar,*Finite and infinite cyclic extensions of free groups*, J. Austral. Math. Soc.**16**(1973), 458–466. Collection of articles dedicated to the memory of Hanna Neumann, IV. MR**0349850****6.**Abe Karrass, Alfred Pietrowski, and Donald Solitar,*Automorphisms of a free product with an amalgamated subgroup*, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 328–340. MR**767119**, 10.1090/conm/033/767119**7.**Sava Krstić,*Finitely generated virtually free groups have finitely presented automorphism group*, Proc. London Math. Soc. (3)**64**(1992), no. 1, 49–69. MR**1132854**, 10.1112/plms/s3-64.1.49**8.**Sava Krstić and Karen Vogtmann,*Equivariant outer space and automorphisms of free-by-finite groups*, Comment. Math. Helv.**68**(1993), no. 2, 216–262. MR**1214230**, 10.1007/BF02565817**9.**Roger C. Lyndon and Paul E. Schupp,*Combinatorial group theory*, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR**0577064****10.**J. McCool,*The automorphism groups of finite extensions of free groups*, Bull. London Math. Soc.**20**(1988), no. 2, 131–135. MR**924240**, 10.1112/blms/20.2.131**11.**Martin R. Pettet,*Finitely generated groups with virtually free automorphism groups*, Proc. Edinburgh Math. Soc. (2)**38**(1995), no. 3, 475–484. MR**1357644**, 10.1017/S0013091500019271**12.**Daniel Segal,*Polycyclic groups*, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR**713786****13.**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
20F28,
20E36,
20E08

Retrieve articles in all journals with MSC (1991): 20F28, 20E36, 20E08

Additional Information

**Martin R. Pettet**

Affiliation:
Department of Mathematics, University of Toledo, Toledo, Ohio 43606

Email:
mpettet@math.utoledo.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01699-1

Received by editor(s):
November 4, 1994

Received by editor(s) in revised form:
January 4, 1966

Article copyright:
© Copyright 1997
American Mathematical Society