The automorphism group of a free group is not a $\textrm {CAT}(0)$ group
Author:
S. M. Gersten
Journal:
Proc. Amer. Math. Soc. 121 (1994), 999-1002
MSC:
Primary 20F32; Secondary 20E05, 20F28, 53C23, 57M07
DOI:
https://doi.org/10.1090/S0002-9939-1994-1195719-9
MathSciNet review:
1195719
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Abstract | References | Similar Articles | Additional Information
Abstract: If F is a finitely generated free group, then the group ${\operatorname {Aut}}(F)$, if ${\text {rank}}(F) \geq 3$, and ${\text {Out}}(F)$, if ${\text {rank}}(F) \geq 4$, are not isomorphic to a subgroup of a group which acts properly discontinuously and cocompactly on a 1-connected geodesic metric space satisfying Gromov’s condition ${\text {CAT}}(0)$.
- Martin R. Bridson, Geodesics and curvature in metric simplicial complexes, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 373–463. MR 1170372 M. R. Bridson, A generalized flat torus theorem with applications (in preparation).
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226
- David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694
- Edward Formanek and Claudio Procesi, The automorphism group of a free group is not linear, J. Algebra 149 (1992), no. 2, 494–499. MR 1172442, DOI https://doi.org/10.1016/0021-8693%2892%2990029-L
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI https://doi.org/10.1007/978-1-4613-9586-7_3
- É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648
- Detlef Gromoll and Joseph A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545–552. MR 281122, DOI https://doi.org/10.1090/S0002-9904-1971-12747-7
- H. Blaine Lawson Jr. and Shing Tung Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211–228. MR 334083
- Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
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Additional Information
Keywords:
Free group,
automorphism,
<!– MATH ${\text {CAT}}(0)$ –> <IMG WIDTH="73" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${\text {CAT}}(0)$">,
geodesic metric space
Article copyright:
© Copyright 1994
American Mathematical Society
