Cyclically presented groups embedded in one-relator products of cyclic groups
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- by Colin M. Campbell, Patricia M. Heggie, Edmund F. Robertson and Richard M. Thomas PDF
- Proc. Amer. Math. Soc. 118 (1993), 401-408 Request permission
Abstract:
We consider the groups defined by the presentations \[ \langle a,b:{a^2} = {b^n} = a{b^{ - 1}}ab{(aba{b^{ - 1}})^{\alpha - 1}}a{b^2}a{b^{ - 2}} = 1\rangle \] and investigate their structure for small values of $\alpha$. This forms part of a general investigation into the structure of groups defined by presentations of the form \[ \langle a,b:{a^2} = {b^n} = w(a,b) = 1\rangle .\] Connections between these groups and the Fibonacci groups are also explored.References
- Gilbert Baumslag, John W. Morgan, and Peter B. Shalen, Generalized triangle groups, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 25–31. MR 886432, DOI 10.1017/S0305004100067013
- A. M. Brunner, The determination of Fibonacci groups, Bull. Austral. Math. Soc. 11 (1974), 11–14. MR 349849, DOI 10.1017/S0004972700043574
- C. M. Campbell, H. S. M. Coxeter, and E. F. Robertson, Some families of finite groups having two generators and two relations, Proc. Roy. Soc. London Ser. A 357 (1977), no. 1691, 423–438. MR 460470, DOI 10.1098/rspa.1977.0177
- C. M. Campbell, P. M. Heggie, E. F. Robertson, and R. M. Thomas, One-relator products of cyclic groups and Fibonacci-like sequences, Applications of Fibonacci numbers, Vol. 4 (Winston-Salem, NC, 1990) Kluwer Acad. Publ., Dordrecht, 1991, pp. 63–68. MR 1193701 —, Finite one-relator products of two cyclic groups with the relator of arbitrary length, J. Austral. Math. Soc. (to appear).
- Colin M. Campbell, Edmund F. Robertson, and Richard M. Thomas, Fibonacci numbers and groups, Applications of Fibonacci numbers (San Jose, CA, 1986) Kluwer Acad. Publ., Dordrecht, 1988, pp. 45–60. MR 951905, DOI 10.1007/978-94-015-7801-1_{6}
- C. M. Campbell, E. F. Robertson, and R. M. Thomas, On groups related to Fibonacci groups, Group theory (Singapore, 1987) de Gruyter, Berlin, 1989, pp. 323–331. MR 981851
- C. M. Campbell and R. M. Thomas, On $(2,n)$-groups related to Fibonacci groups, Israel J. Math. 58 (1987), no. 3, 370–380. MR 917365, DOI 10.1007/BF02771698
- C. P. Chalk and D. L. Johnson, The Fibonacci groups. II, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 1-2, 79–86. MR 463301, DOI 10.1017/S0308210500018059
- Marston Conder, Three-relator quotients of the modular group, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 427–447. MR 916226, DOI 10.1093/qmath/38.4.427 J. H. Conway, Advanced problem $5327$, Amer. Math. Monthly 72 (1965), 915. J. H. Conway et al., Solution to advanced problem $5327$, Amer. Math. Monthly 74 (1967), 91-93. H. Doostie, Fibonacci-type sequences and classes of groups, PhD Thesis, University of St Andrews, 1988.
- Benjamin Fine, James Howie, and Gerhard Rosenberger, One-relator quotients and free products of cyclics, Proc. Amer. Math. Soc. 102 (1988), no. 2, 249–254. MR 920981, DOI 10.1090/S0002-9939-1988-0920981-1
- Benjamin Fine, James Howie, and Gerhard Rosenberger, Ree-Mendelsohn pairs in generalized triangle groups, Comm. Algebra 17 (1989), no. 2, 251–258. MR 978473, DOI 10.1080/00927878908823726
- Benjamin Fine, Frank Levin, and Gerhard Rosenberger, Free subgroups and decompositions of one-relator products of cyclics. I. The Tits alternative, Arch. Math. (Basel) 50 (1988), no. 2, 97–109. MR 930108, DOI 10.1007/BF01194564 —, Free subgroups and decompositions of one relator products of cyclics. Part 2: Normal torsion-free subgroups and fpa decompositions, J. Indian Math. Soc. 49 (1988), 237-247.
- B. Fine and G. Rosenberger, A note on generalized triangle groups, Abh. Math. Sem. Univ. Hamburg 56 (1986), 233–244. MR 882417, DOI 10.1007/BF02941518
- Benjamin Fine and Gerhard Rosenberger, Complex representations and one-relator products of cyclics, Geometry of group representations (Boulder, CO, 1987) Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 131–147. MR 957516, DOI 10.1090/conm/074/957516
- George Havas, A Reidemeister-Schreier program, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 347–356. MR 0376827
- George Havas, Computer aided determination of a Fibonacci group, Bull. Austral. Math. Soc. 15 (1976), no. 2, 297–305. MR 424949, DOI 10.1017/S0004972700022668
- George Havas, P. E. Kenne, J. S. Richardson, and E. F. Robertson, A Tietze transformation program, Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 69–73. MR 760651 H. Helling, A. C. Kim, and J. L. Mennicke, On Fibonacci groups, preprint.
- Gabriele Kern-Isberner and Gerhard Rosenberger, Normalteiler vom Geschlecht eins in freien Produkten endlicher zyklischer Gruppen, Results Math. 11 (1987), no. 3-4, 272–288 (German, with English summary). MR 897303, DOI 10.1007/BF03323275 R. C. Lyndon, On a family of infinite groups introduced by Conway, unpublished.
- M. F. Newman, Proving a group infinite, Arch. Math. (Basel) 54 (1990), no. 3, 209–211. MR 1037607, DOI 10.1007/BF01188513 M. F. Newman and E. A. O’Brien, A computer-aided analysis of some finitely-presented groups, J. Austral. Math. Soc. (to appear).
- Stephen J. Pride, Groups with presentations in which each defining relator involves exactly two generators, J. London Math. Soc. (2) 36 (1987), no. 2, 245–256. MR 906146, DOI 10.1112/jlms/s2-36.2.245
- Richard M. Thomas, The Fibonacci groups $F(2,2m)$, Bull. London Math. Soc. 21 (1989), no. 5, 463–465. MR 1005823, DOI 10.1112/blms/21.5.463
- Richard M. Thomas, The Fibonacci groups revisited, Groups—St. Andrews 1989, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 160, Cambridge Univ. Press, Cambridge, 1991, pp. 445–454. MR 1123998, DOI 10.1017/CBO9780511661846.017
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 401-408
- MSC: Primary 20F05
- DOI: https://doi.org/10.1090/S0002-9939-1993-1140665-9
- MathSciNet review: 1140665