# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Gilbert Baumslag, Some problems on one-relator groups, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Springer, Berlin, 1974, pp. 75 – 81. Lecture Notes in Math., Vol. 372. [2] G. Baumslag, M. R. Bridson, C. F. Miller III, H. Short, Subgroups of Automatic Groups, pre-print. [3] G. Baumslag, C. F. Miller III, and H. Short, Unsolvable problems about small cancellation and word hyperbolic groups, Bull. London Math. Soc. 26 (1994), no. 1, 97 – 101. · Zbl 0810.20025 · doi:10.1112/blms/26.1.97 · doi.org [4] Gilbert Baumslag and James E. Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), no. 1, 44 – 52. · Zbl 0559.20018 · doi:10.1112/jlms/s2-30.1.44 · doi.org [5] Robert Bieri, Homological dimension of discrete groups, Mathematics Department, Queen Mary College, London, 1976. Queen Mary College Mathematics Notes. · Zbl 0357.20027 [6] M. R. Bridson and A. Haefliger, Metric spaces of nonpositive curvature, to appear. · Zbl 0988.53001 [7] S. M. Gersten, Questions on Geometric Group Theory for the Max Dehn Seminar, available by ftp at: ftp.math.utah.edu /u/ma/gersten/MaxDehnSeminar (1995). [8] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75 – 263. · Zbl 0634.20015 · doi:10.1007/978-1-4613-9586-7_3 · doi.org [9] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. · Zbl 0368.20023 [10] E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45 – 47. · Zbl 0481.20020 · doi:10.1112/blms/14.1.45 · doi.org [11] G. P. Scott, Finitely generated 3-manifold groups are finitely presented, J. London Math. Soc. (2) 6 (1973), 437 – 440. · Zbl 0254.57003 · doi:10.1112/jlms/s2-6.3.437 · doi.org [12] John Cossey , Problem section, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Springer, Berlin, 1974, pp. 733 – 740. Lecture Notes in Math., Vol. 372.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.