A quick test for nonisomorphism of one-relator groups

Author:
K. J. Horadam

Journal:
Proc. Amer. Math. Soc. **81** (1981), 195-200

MSC:
Primary 20F05; Secondary 03D40, 20F10, 20J05

DOI:
https://doi.org/10.1090/S0002-9939-1981-0593456-3

MathSciNet review:
593456

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Abstract: A sequence of integers is read off from the presentation of a finitely generated torsion-free one-relator group with nontrivial second integral homology, without recourse to group-theoretic manipulations. This test sequence is derived from the cup coproduct on the coring of the integral homology module of the group, and reflects information about the group's second lower central factor group.

Test sequences differ only if the corresponding groups are nonisomorphic. The test process can be generalised to any one-relator group with nontrivial second integral homology.

**[1]**G. Baumslag,*Some problems on one-relator groups*(Proc. Second Internat. Conf. Theory of Groups, Canberra, 1973), Lecture Notes in Math., Vol. 372, Springer-Verlag, Berlin, 1974, pp. 75-81. MR**0364463 (51:717)****[2]**R. H. Fox,*Free differential calculus*. I, Ann. of Math. (2)**57**(1953), 547-560. MR**0053938 (14:843d)****[3]**K. J. Horadam,*The diagonal comultiplication on homology*, J. Pure Appl. Algebra (to appear). MR**601682 (82m:20053)****[4]**A. G. Kurosh,*The theory of groups*, Vol. II, Chelsea, New York, 1956. MR**0080089 (18:188f)****[5]**M. Newman,*Integral matrices*, Academic Press, New York, 1972. MR**0340283 (49:5038)****[6]**W. Magnus, A. Karrass and D. Solitar,*Combinatorial group theory*, 2nd rev. ed., Dover, New York, 1976. MR**0422434 (54:10423)****[7]**S. J. Pride,*The isomorphism problem for two-generator one-relator groups with torsion is solvable*, Trans. Amer. Math. Soc.**227**(1977), 109-139. MR**0430085 (55:3092)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0593456-3

Keywords:
One-relator group,
isomorphism problem,
lower central sequence,
integral homology module,
cup coproduct,
Fox derivative,
invariant factors of matrix

Article copyright:
© Copyright 1981
American Mathematical Society