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

Full-text PDF

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]**Michael Anshel and Peter Stebe,*The solvability of the conjugacy problem for certain HNN groups*, Bull. Amer. Math. Soc.**80**(1974), 266–270. MR**0419615**, https://doi.org/10.1090/S0002-9904-1974-13455-5**[2]**J. Barwise (ed.),*Handbook of mathematical logic*, North-Holland, Amsterdam, 1977. MR**0457132 (56:15351)****[3]**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**0409666**, https://doi.org/10.1007/BF01301574**[4]**Rudolf Halin,*Graphen ohne unendliche Wege*, Math. Nachr.**31**(1966), 111–123 (German). MR**0191838**, https://doi.org/10.1002/mana.19660310110**[5]**Philip J. Higgins,*Notes on categories and groupoids*, Van Nostrand Reinhold Co., London-New York-Melbourne, 1971. Van Nostrand Rienhold Mathematical Studies, No. 32. MR**0327946****[6]**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**, https://doi.org/10.1090/S0002-9939-1981-0609643-1**[7]**R. Daniel Hurwitz,*On cyclic subgroups and the conjugacy problem*, Proc. Amer. Math. Soc.**79**(1980), no. 1, 1–8. MR**560573**, https://doi.org/10.1090/S0002-9939-1980-0560573-2**[8]**Leif Larsen,*The conjugacy problem and cyclic HNN constructions*, J. Austral. Math. Soc. Ser. A**23**(1977), no. 4, 385–401. MR**0480757****[9]**Seymour Lipschutz,*The conjugacy problem and cyclic amalgamations*, Bull. Amer. Math. Soc.**81**(1975), 114–116. MR**0379675**, https://doi.org/10.1090/S0002-9904-1975-13661-5**[10]**Charles F. Miller III,*On group-theoretic decision problems and their classification*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 68. MR**0310044****[11]**O. Ore,*Theory of graphs*, Amer. Math. Soc. Colloq. Publ., vol. 38, Amer. Math. Soc., Providence, R.I., 1967.**[12]**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20F10,
20L10

Retrieve articles in all journals with MSC: 20F10, 20L10

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