Algorithms and geometry for graph products of groups. (English) Zbl 0831.20032

Hermiller, Susan; Meier, John

J. Algebra 171, No. 1, 230-257 (1995).

Given a finite simplicial graph \(\Gamma\) with a group \(G_v\) for each vertex \(v\) of \(\Gamma\) the associated graph product \(G(\Gamma)\) is the group generated by each of the vertex groups with the added relations that elements of distinct adjacent vertex groups commute. If the graph consists only of vertices, the graph product is just the free product of the vertex groups. If the graph is complete, that is any two vertices are joined by an edge, the graph product is the direct product. Right angled Coxeter groups \(C(\Gamma)\) (respectively right angled Artin groups \(A(\Gamma))\) are graph products where each vertex group is cyclic of order 2 (respectively infinite cyclic).
It is natural to ask which group properties are preserved under taking finite graph products. The authors show that being semi-hyperbolic, being automatic and admitting a complete rewriting system are all properties preserved under forming finite graph products.

Reviewer: J.Harlander (Frankfurt am Main)

MSC:

20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05	Generators, relations, and presentations of groups
20F65	Geometric group theory
20F10	Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
05C25	Graphs and abstract algebra (groups, rings, fields, etc.)

Keywords:

right angled Coxeter groups; semi-hyperbolic groups; automatic groups; finite simplicial graphs; vertex groups; relations; free products; direct products; right angled Artin groups; graph products; complete rewriting systems