zbMATH — the first resource for mathematics

Decomposition of a group as free product or amalgamated sum. (Décomposition d’un groupe en produit libre ou somme amalgamée.) (French) Zbl 0836.20038
The Grusko theorem asserts that the rank (minimal number of generators) is additive with respect to a free product. The relations are subject to an analogous property. Let \(G\) be a finitely presented group; one defines \(T(G)\) as the minimal number of relations of a triangular presentation of \(G\), i.e., a presentation where relators have length 3. The main result is that, like the rank, \(T\) is additive: \(T(A * B) = T(A) +T(B)\). A relative invariant \(T(G, C)\) is defined for a pair consisting of a group \(G\) with a subgroup \(C\). Under some hypothesis, \(T(A *_C B) \geq T(A,C) + (B,C)\). Several applications of this inequality are given.

20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Full Text: DOI Crelle EuDML