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When is the graph product of hyperbolic groups hyperbolic?

✍ Scribed by John Meier

Publisher: Springer
Year: 1996
Tongue: English
Weight: 771 KB
Volume: 61
Category: Article
ISSN: 0046-5755
DOI: 10.1007/bf00149417

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✦ Synopsis

Given a finite simplicial graph G, and an assignment of groups to the vertices of if, the graph product is the free product of the vertex groups modulo relations implying that adjacent vertex groups commute. We use Gromov's link criteria for cubical complexes and techniques of Davis and Moussang to study the curvature of graph products of groups. By constructing a CAT(-1) cubical complex, it is shown that the graph product of word hyperbolic groups is itself word hyperbolic if and only if the full subgraph ~a in G, generated by vertices whose associated groups are finite, satisfies three specific criteria. The construction shows that arbitrary graph products of finite groups are Bridson groups.

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