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Commensurability of graph products. (English) Zbl 0998.20029

Two groups \(G\), \(G^*\) are commensurable if there is a group \(H\) isomorphic to a subgroup of finite index in both \(G\) and \(G^*\); they are strongly commensurable if \(H\) has the same index in both \(G\) and \(G^*\). The authors prove that, if \((G_v)_{v\in V}\), \((G^*_v)_{v\in V}\) are two families of groups indexed by vertices of a finite graph and if, for each \(v\in V\), \(G_v\) and \(G_v^*\) are strongly commensurable, then the two corresponding graph products are also strongly commensurable.
The proof involves two complementary descriptions of right-angled buildings on which a graph product acts.
Since a group commensurable with a linear group is linear, the result allows the authors to prove the linearity of a number of graph products whose linearity was not known before.
The authors generalize the main result somewhat to allow commensurability, under certain circumstances, to replace strong commensurability; but examples show that additional hypotheses are required.
Reviewer: J.W.Cannon (Provo)

MSC:

20E42 Groups with a \(BN\)-pair; buildings
20F65 Geometric group theory
57M07 Topological methods in group theory
20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20E08 Groups acting on trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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