Higher Generation Subgroup Sets and the Virtual Cohomological Dimension of Graph Products of Finite Groups |
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Authors: | Harlander Jens; Meinert Holger |
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Institution: | Fachbereich Mathematik, Johann Wolfgang Goethe-Universität Robert-Mayer-Str. 6-10, 60054 Frankfurt a.M., Germany |
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Abstract: | We introduce panels of stabilizer schemes (K, G*) associatedwith finite intersection-closed subgroup sets of a given groupG, generalizing in some sense Davis' notion of a panel structureon a triangulated manifold for Coxeter groups. Given (K, G*),we construct a G-complex X with K as a strong fundamental domainand simplex stabilizers conjugate to subgroups in . It turnsout that higher generation properties of in the sense of Abels-Holzare reflected in connectivity properties of X. Given a finite simplicial graph and a non-trivial group G()for every vertex of , the graph product G() is the quotientof the free product of all vertex groups modulo the normal closureof all commutators G(), G(w)] for which the vertices , w areadjacent. Our main result allows the computation of the virtualcohomological dimension of a graph product with finite vertexgroups in terms of connectivity properties of the underlyinggraph . |
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