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Inversion is Possible in Groups with no Periodic Automorphisms

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  05 June 2015

Martin R. Bridson  and
Hamish Short
Martin R. Bridson
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford OX2 6GG, UK ([email protected])
Hamish Short
Affiliation:
Aix Marseille Université, Centre National de la Recherche Scientifique, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France ([email protected])
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Abstract

There exist infinite finitely presented torsion-free groups G such that Aut(G) and Out(G) are torsion free but G has an automorphism sending some non-trivial element to its inverse.

Keywords

group automorphismshyperbolic groupscomplete groups

MSC classification

Secondary: 20E36: Automorphisms of infinite groups 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 11 - 16
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Alonso, J., Brady, T., Cooper, D., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M. and Short, H., Notes on word hyperbolic groups, in Group theory from a geometrical viewpoint (ed. Ghys, E., Haefliger, A. and Verjovsky, A.), pp. 3–63 (World Scientific, 1991).Google Scholar
2. Belolipetsky, M. and Lubotzky, A., Finite groups and hyperbolic manifolds, Invent. Math. 162 (2005), 459–472.Google Scholar
3. Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften, Volume 319 (Springer, 1999).Google Scholar
4. Bridson, M. R. and Short, H., A complete embedding theorem for torsion-free groups, in preparation.Google Scholar
5. Gromov, M., Hyperbolic groups, in Essays in group theory (ed. Gersten, S. M.), Mathematical Sciences Research Institute Publications, Volume 8, pp. 75–263 (Springer, 1987).CrossRefGoogle Scholar
6. Hegarty, P. and MacHale, D., Minimal odd order automorphism groups, J. Group Theory 13 (2010), 243–255.CrossRefGoogle Scholar
7. Heineken, H. and Liebeck, H., On p-groups with odd order automorphism groups, Arch. Math. 24 (1973), 464–471.CrossRefGoogle Scholar
8. Kojima, S., Isometry transformations of hyperbolic 3-manifolds, Topol. Applic. 29 (1988), 297–307.CrossRefGoogle Scholar
9. Long, D. D. and Reid, A. W., On asymmetric hyperbolic manifolds, Proc. Camb. Phil. Soc. 138 (2005), 301–306.Google Scholar
10. Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory: presentations of groups in terms of generators and relations (Interscience Publishers, New York, 1966).Google Scholar
11. Mihalik, M. L. and Towle, W., Quasiconvex subgroups of negatively curved groups, J. Pure Appl. Alg. 95 (1994), 297–301.CrossRefGoogle Scholar
12. C. F., Miller III and Schupp, P. E., Embedding into Hopfian groups, J. Alg. 17 (1971), 171–176.Google Scholar
13. Sela, Z., Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups, II, Geom. Funct. Analysis 7 (1997), 561–593.CrossRefGoogle Scholar