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Amalgamated sums of groups

Published online by Cambridge University Press:  20 January 2009

J. M. Corson
J. M. Corson
Affiliation:
Department of MathematicsUniversity of AlabamaBox 870350Tuscaloosa, AL 35487–0350U.S.A. E-mail address: jcorson@mathdept.as.ua.edu
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Abstract

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Groups called amalgamated sums that arise as inductive limits of systems of groups and injective homomorphisms are studied. The problem is to find conditions under which the groups in the system do not collapse in the limit. Such a condition is given by J. Tits when certain subsystems are associated to buildings. This condition can be phrased to apply to certain systems of abstract groups and injective homomorphisms. It is shown to imply that no collapse occurs in the limit in a strong sense; namely the natural homomorphism of the amalgamated sum of any subsystem into the amalgamated sum of the full system is injective. This answers a question of S. J. Pride.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 3 , October 1996 , pp. 561 - 570
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Bogley, W. and Pride, S. J., Calculating generators of π2, in Two-dimensional homotopy and combinatorial group theory (Hog-Angeloni, C., Metzler, W., Sierodsky, A., eds., London Mathematical Society Lecture Note Series 197, Cambridge University Press, Cambridge, 1993), 157–188.CrossRefGoogle Scholar
2. Chermak, A., Non-spherical assemblages and hyperbolic automorphisms of trees, preprint.Google Scholar
3. Corson, J. M., Complexes of groups, Proc. London Math. Soc. (3) 65 (1992), 199–224.CrossRefGoogle Scholar
4. Edjvet, M., On a certain class of group presentations, Math. Proc. Cambridge Philos. Soc. 105 (1989), 25–35.CrossRefGoogle Scholar
5. Haefliger, A., Complexes of groups and orbihedra, in Group theory from a geometrical viewpoint (Ghys, , Haefliger, and Verjovsky, , eds., World Scientific, Singapore, 1991), 504–540.Google Scholar
6. Lyndon, R. C. and Schupp, P. E., Combinatorial group theory (Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, Berlin-Heidelberg-New York, 1977).Google Scholar
7. Pride, S. J., Groups with presentations in which each defining relator involves exactly two generators, J. London Math. Soc. (2) 36 (1987), 245–256.CrossRefGoogle Scholar
8. Pride, S. J., The (co)homology of groups given by presentations in which each defining relator involves at most two types of generators, J. Austral. Math. Soc. Ser. A 52 (1992), 205–218.CrossRefGoogle Scholar
9. Spieler, B., Ph.D. thesis, The Ohio State University, 1992.Google Scholar
10. Stallings, J. R., Non-positively curved triangles of groups, in Group theory from a geometrical viewpoint (Ghys, , Haefliger, and Verjovsky, , eds, World Scientific, Singapore, 1991), 491–503.Google Scholar
11. Tits, J., Buildings and group amalgamations, in Proceedings of Groups—St Andrews 1985 (Robertson, E. F. and Campbell, C. M., eds., London Mathematical Society Lecture Notes Series 121, Cambridge University Press, Cambridge, 1986), 110–127.Google Scholar