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Incoherent negatively curved groups. (English) Zbl 0913.20027
A group is said to be coherent if every finitely generated subgroup is finitely presented. E. Rips [Bull. Lond. Math. Soc. 14, 45-47 (1982; Zbl 0481.20020)] answered negatively the question of whether \(C(8)\) small-cancellation groups are coherent or not. Rips’ result is based on a construction which provides a short exact sequence \(1\to N\to G\to Q\to 1\), where \(G\) is a finitely presented small-cancellation group and \(N\) a finitely generated normal subgroup generated by two elements of \(G\).
In this paper, the author answers negatively a question of Gersten, who asked whether the fundamental groups of negatively curved 2-complexes are coherent. The paper has two parts. In Part 1, the author provides a construction similar to that of Rips, but in which \(G\) is the fundamental group of a compact negatively curved 2-complex instead of a small-cancellation group. Applications of Rips’ construction are also extended, and the author constructs his first examples of incoherent negatively curved groups which answer Gersten’s question. In Part 2, the author provides a different method, which is more geometric, for constructing incoherent finitely presented groups, and using this method he gives an easy new example of a compact negatively curved 2-complex with incoherent fundamental group.

MSC:
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
57M05 Fundamental group, presentations, free differential calculus
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