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• W1970945686 abstract "Finite graph products of groups with solvable conjugacy problem and with finite cyclic edge groups are considered. It is shown that the graph product has solvable conjugacy problem if each of the vertex groups has solvable conjugacy problem. The conjugacy problem for graph products (the fundamental group of a graph of groups as defined by Bass and Serre; see Serre [2]) was studied by Horadam [1]. Horadam showed that finite graph products with finite cyclic edge groups inherit a solvable conjugacy problem from their vertex groups under certain conditions, one of which is that the images of the edge group generators in each vertex group are powers of a common central element. We show that this condition is not necessary and, in fact, that all finite graph products with finite cyclic edge groups inherit a solvable conjugacy problem from their vertex groups. Let D be a finite connected directed graph with edge set E, vertex set V, and source and terminus maps s and t, respectively. Let (G, D) be a graph of groups; that is, for each v E V, there is a vertex group Gv and for each e E E, there is an edge group Ge . Further, for each e E E, there are partially recursive monomorphisms Ae: Ge Gse and Be: Ge -* Gte . Let T be a maximal tree of D and let E' be the edge set of T. Then the graph product G* has presentation G= (Gv (v E V), e (e EE); eBe(x)e= Ae(x) (x E G, e E E), e = 1 (e E E')). Note that G* is a finite sequence of HNN-extensions (one for each e in E-E') of a tree product G*=(Gv (vEV); Ae(x)=Be(x)(xEGe, eEF')), which is itself a finite sequence of free products with amalgamation (one for each e in E'). We will consider graph products such that for each e E E, Ge is a finite cyclic group (kie). Received by the editors July 10, 1990 and, in revised form, July 16, 1991. 1991 Mathematics Subject Classification. Primary 20E06, 20F10; Secondary 20F05. Research supported by a grant from the Naval Academy Research Council ( 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page" @default.
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• W1970945686 date "1992-03-01" @default.
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• W1970945686 title "The conjugacy problem for graph products with infinite cyclic edge groups" @default.
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• W1970945686 doi "https://doi.org/10.1090/s0002-9939-1992-1072088-4" @default.
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