On finite presentations for groups

Craggs, R.

doi:10.1090/S0002-9939-1980-0550487-6

On finite presentations for groups
HTML articles powered by AMS MathViewer

by R. Craggs

Proc. Amer. Math. Soc. 78 (1980), 170-174

DOI: https://doi.org/10.1090/S0002-9939-1980-0550487-6

PDF | Request permission

Abstract:

Three theorems are proved here giving canonical forms for finitely-presented groups. The first characterizes groups with balanced presentations as those groups with presentations of the form \[ \langle {y_1}, \ldots ,{y_n}|{y_i} = \alpha ({y_i}),i = 1, \ldots ,n\rangle \] where $\alpha$ is some automorphism of the free group with free basis ${y_1}, \ldots ,{y_n}$. Similar canonical forms, with two automorphisms rather than one, are given for arbitrary finitely-presented groups.

References

S. I. Adyan, On algorithmic problems in effectively complete classes of groups, Dokl. Akad. Nauk SSSR 123 (1958), 13–16 (Russian). MR 0103217
J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 173241, DOI 10.1090/S0002-9939-1965-0173241-8
J. J. Andrews and M. L. Curtis, Extended Nielsen operations in free groups, Amer. Math. Monthly 73 (1966), 21–28. MR 195928, DOI 10.2307/2313917
W. W. Boone, Decision problems about algebraic and logical systems as a whole and recursively enumerable degrees of unsolvability, Contributions to Math. Logic (Colloquium, Hannover, 1966) North-Holland, Amsterdam, 1968, pp. 13–33. MR 0237617

dimensional open books

I. Gruschko, Über die Basen eines freien Produktes von Gruppen, Rec. Math. [Mat. Sbornik] N.S. 8 (50) (1940), 169–182 (Russian, with German summary). MR 0003412
G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR 130286, DOI 10.1098/rspa.1961.0132
Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 32641, DOI 10.1112/jlms/s1-24.4.247

The theory of groups

Roger C. Lyndon, Grushko’s theorem, Proc. Amer. Math. Soc. 16 (1965), 822–826. MR 178073, DOI 10.1090/S0002-9939-1965-0178073-2
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3

Combinatorial group theory

Presentations of groups in terms of generators and relations

Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich

44

William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390
Wolfgang Metzler, Über den Homotopietyp zweidimensionaler $CW$-Komplexe und Elementartransformationen bei Darstellungen von Gruppen durch Erzeugende und definierende Relationen, J. Reine Angew. Math. 285 (1976), 7–23. MR 440527, DOI 10.1515/crll.1976.285.7
Charles F. Miller III, On group-theoretic decision problems and their classification, Annals of Mathematics Studies, No. 68, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0310044
Robert Myers, Open book decompositions of $3$-manifolds, Proc. Amer. Math. Soc. 72 (1978), no. 2, 397–402. MR 507346, DOI 10.1090/S0002-9939-1978-0507346-5
B. H. Neumann, On the number of generators of a free product, J. London Math. Soc. 18 (1943), 12–20. MR 8809, DOI 10.1112/jlms/s1-18.1.12
Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), no. 1, 189–358 (German). MR 1555256, DOI 10.1007/BF02421324
Renée Peiffer, Über Identitäten zwischen Relationen, Math. Ann. 121 (1949), 67–99 (German). MR 32640, DOI 10.1007/BF01329617
Michael O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172–194. MR 110743, DOI 10.2307/1969933
Elvira Strasser Rapaport, Note on Nielsen transformations, Proc. Amer. Math. Soc. 10 (1959), 228–235. MR 104724, DOI 10.1090/S0002-9939-1959-0104724-1
Elvira Strasser Rapaport, Groups of order $1$: Some properties of presentations, Acta Math. 121 (1968), 127–150. MR 229704, DOI 10.1007/BF02391911
John R. Stallings, A topological proof of Grushko’s theorem on free products, Math. Z. 90 (1965), 1–8. MR 188284, DOI 10.1007/BF01112046
Perrin Wright, Group presentations and formal deformations, Trans. Amer. Math. Soc. 208 (1975), 161–169. MR 380813, DOI 10.1090/S0002-9947-1975-0380813-5