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The automorphism group of a free group is not a $ {\rm CAT}(0)$ group

Author: S. M. Gersten
Journal: Proc. Amer. Math. Soc. 121 (1994), 999-1002
MSC: Primary 20F32; Secondary 20E05, 20F28, 53C23, 57M07
MathSciNet review: 1195719
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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)$.

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  • [Br1] M. R. Bridson, Geodesics and curvature in metric simplicial complexes, Group Theory from a Geometrical Viewpoint (E. Ghys, A. Haefliger, and A. Verjovsky, eds.), World Scientific, 1990, pp. 373-463. MR 1170372 (94c:57040)
  • [Br2] M. R. Bridson, A generalized flat torus theorem with applications (in preparation).
  • [BF] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, Differential Geom. 35 (1992), 85-101. MR 1152226 (93d:53053)
  • [ECHLPT] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word processing in groups, Bartlett and Jones, Boston, MA, 1992. MR 1161694 (93i:20036)
  • [FP] E. Formanek and C. Procesi, The automorphism group of a free group is not linear, J. Algebra 149 (1992), 494-499. MR 1172442 (93h:20038)
  • [Gr] M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), Math. Sci. Res. Inst. Publ., vol. 8, Springer-Verlag, New York and Berlin, 1987. MR 919829 (89e:20070)
  • [GH] E. Ghys and P. de la Harpe, Notes sur les groupes hyperboliques de Mikhael Gromov, Birkhäuser, Boston, MA, 1990. MR 1086648 (92f:53050)
  • [GW] D. Gromoll and J. 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 0281122 (43:6841)
  • [LY] H. B. Lawson and S. T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geom. 7 (1972), 211-228. MR 0334083 (48:12402)
  • [Se] J.-P. Serre, Trees, Springer-Verlag, New York, 1980. MR 607504 (82c:20083)

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Additional Information

Keywords: Free group, automorphism, $ {\text{CAT}}(0)$, geodesic metric space
Article copyright: © Copyright 1994 American Mathematical Society

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