The isomorphism problem for two-generator one-relator groups with torsion is solvable

Pride, Stephen J.

doi:10.1090/S0002-9947-1977-0430085-X

The isomorphism problem for two-generator one-relator groups with torsion is solvable
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by Stephen J. Pride PDF

Trans. Amer. Math. Soc. 227 (1977), 109-139 Request permission

Abstract:

The theorem stated in the title is obtained by determining (in a sense to be made precise) all the generating pairs of an arbitrary two-generator one-relator group with torsion. As a consequence of this determination it is also deduced that every two-generator one-relator group G with torsion is Hopfian, and that the automorphism group of G is finitely generated.

References

Donald J. Collins, Recursively enumerable degrees and the conjugacy problem, Acta Math. 122 (1969), 115–160. MR 242671, DOI 10.1007/BF02392008
G. A. Gurevič, On the conjugacy problem for groups with one defining relation, Dokl. Akad. Nauk SSSR 207 (1972), 18–20 (Russian). MR 0316576
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
James McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975), 205–213. MR 396764, DOI 10.1016/0021-8693(75)90045-9
James McCool and Paul E. Schupp, On one relator groups and $\textrm {HNN}$ extensions, J. Austral. Math. Soc. 16 (1973), 249–256. Collection of articles dedicated to the memory of Hanna Neumann, II. MR 0338186, DOI 10.1017/S1446788700014300
D. I. Moldavanskiĭ, Certain subgroups of groups with one defining relation, Sibirsk. Mat. Ž. 8 (1967), 1370–1384 (Russian). MR 0220810
B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968), 568–571. MR 222152, DOI 10.1090/S0002-9904-1968-12012-9
B. B. Newman, The soluble subgroups of a one-relator group with torsion, J. Austral. Math. Soc. 16 (1973), 278–285. Collection of articles dedicated to the memory of Hanna Neumann, III. MR 0338188, DOI 10.1017/S1446788700015044
Stephen J. Pride, On the generation of one-relator groups, Trans. Amer. Math. Soc. 210 (1975), 331–364. MR 382458, DOI 10.1090/S0002-9947-1975-0382458-X
Stephen J. Pride, Certain subgroups of certain one-relator groups, Math. Z. 146 (1976), no. 1, 1–6. MR 393253, DOI 10.1007/BF01213712
George S. Sacerdote and Paul E. Schupp, SQ-universality in HNN groups and one relator groups, J. London Math. Soc. (2) 7 (1974), 733–740. MR 364464, DOI 10.1112/jlms/s2-7.4.733