The conjugacy problem for graph products with cyclic edge groups

Horadam, K. J.

doi:10.1090/S0002-9939-1983-0684622-9

The conjugacy problem for graph products with cyclic edge groups
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Proc. Amer. Math. Soc. 87 (1983), 379-385 Request permission

Abstract:

A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN groups and free products with amalgamation. The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical. For graph products over infinite graphs these conditions are insufficient: a further condition ensures that graph products over infinite graphs of bounded path length have solvable conjugacy problem. These results generalise the known ones for HNN groups and free products with amalgamation.

References

Michael Anshel and Peter Stebe, The solvability of the conjugacy problem for certain HNN groups, Bull. Amer. Math. Soc. 80 (1974), 266–270. MR 419615, DOI 10.1090/S0002-9904-1974-13455-5
Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. MR 457132
Leo P. Comerford Jr. and Bernard Truffault, The conjugacy problem for free products of sixth-groups with cyclic amalgamation, Math. Z. 149 (1976), no. 2, 169–181. MR 409666, DOI 10.1007/BF01301574
Rudolf Halin, Graphen ohne unendliche Wege, Math. Nachr. 31 (1966), 111–123 (German). MR 191838, DOI 10.1002/mana.19660310110
Philip J. Higgins, Notes on categories and groupoids, Van Nostrand Rienhold Mathematical Studies, No. 32, Van Nostrand Reinhold Co., London-New York-Melbourne, 1971. MR 0327946
K. J. Horadam, The word problem and related results for graph product groups, Proc. Amer. Math. Soc. 82 (1981), no. 2, 157–164. MR 609643, DOI 10.1090/S0002-9939-1981-0609643-1
R. Daniel Hurwitz, On cyclic subgroups and the conjugacy problem, Proc. Amer. Math. Soc. 79 (1980), no. 1, 1–8. MR 560573, DOI 10.1090/S0002-9939-1980-0560573-2
Leif Larsen, The conjugacy problem and cyclic HNN constructions, J. Austral. Math. Soc. Ser. A 23 (1977), no. 4, 385–401. MR 480757, DOI 10.1017/s1446788700019546
Seymour Lipschutz, The conjugacy problem and cyclic amalgamations, Bull. Amer. Math. Soc. 81 (1975), 114–116. MR 379675, DOI 10.1090/S0002-9904-1975-13661-5
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

Theory of graphs

Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504