The conjugacy problem for graph products with cyclic edge groups

Author:
K. J. Horadam

Journal:
Proc. Amer. Math. Soc. **87** (1983), 379-385

MSC:
Primary 20F10; Secondary 20L10

DOI:
https://doi.org/10.1090/S0002-9939-1983-0684622-9

MathSciNet review:
684622

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**M. Anshel and P. Stebe,*The solvability of the conjugacy problem for certain*HNN*groups*, Bull. Amer. Math. Soc.**80**(1974), 266-270. MR**0419615 (54:7633)****[2]**J. Barwise (ed.),*Handbook of mathematical logic*, North-Holland, Amsterdam, 1977. MR**0457132 (56:15351)****[3]**L. Comerford and B. Truffault,*The conjugacy problem for free products of sixth-groups with cyclic amalgamation*, Math. Z.**149**(1976), 169-181. MR**0409666 (53:13418)****[4]**R. Halin,*Graphen ohne unendliche Wege*, Math. Nachr**31**(1966), 111-123. MR**0191838 (33:65)****[5]**P. J. Higgins,*Notes on categories and groupoids*, Math. Studies, no. 32, Van Nostrand Reinhold, London, 1971. MR**0327946 (48:6288)****[6]**K. J. Horadam,*The word problem and related results for graph product groups*, Proc. Amer. Math Soc.**82**(1981), 157-164. MR**609643 (82e:20043)****[7]**R. D. Hurwitz,*On cyclic subgroups and the conjugacy problem*, Proc Amer. Math. Soc.**79**(1980), 1-8. MR**560573 (81k:20048)****[8]**L. Larsen,*The conjugacy problem and cyclic*HNN*constructions*, J. Austral. Math. Soc. Ser. A**23**(1977), 385-401. MR**0480757 (58:909)****[9]**S. Lipschutz,*The conjugacy problem and cyclic amalgamations*, Bull. Amer. Math. Soc.**81**(1975). 114-116. MR**0379675 (52:580)****[10]**C. F. Miller III,*On group-theoretic decision problems and their classification*, Ann of Math. Studies, no. 68, Princeton Univ. Press, Princeton, N. J., 1971. MR**0310044 (46:9147)****[11]**O. Ore,*Theory of graphs*, Amer. Math. Soc. Colloq. Publ., vol. 38, Amer. Math. Soc., Providence, R.I., 1967.**[12]**J.-P. Serre,*Trees*(transl. J. Stillwell), Springer-Verlag, Berlin and New York, 1980. MR**607504 (82c:20083)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0684622-9

Keywords:
Groupoid,
graph of groups,
fundamental group,
conjugacy problem

Article copyright:
© Copyright 1983
American Mathematical Society