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Equivalences of two-complexes, with applications to NEC-groups. (English) Zbl 0742.20049

This is a nice paper where the authors study properties of arbitrary 2- complexes and apply them later to the study of non-euclidean crystallographic groups (NEC-groups).
The paper is divided in two parts. In the first one, they introduce the elementary theory of arbitrary 2-complexes, later they discuss equivalences of 2-complexes and give a characterization of equivalences in terms of “Tietze transformations”. Finally they describe the technique called by the authors, “Level Method” which is a method to simplify 2-complexes and they make use of it to show how to simplify certain “quadratic-like” complexes.
In the second part of the paper the authors apply the techniques to obtain the subgroup theorems of A. H. M. Hoare, A. Karrass and D. Solitar [Math. Z. 120, 289-298 (1971; Zbl 0223.20053), Math. Z. 125, 59-69 (1972; Zbl 0223.20054) and Commun. Pure Appl. Math. 26, 731-744 (1973; Zbl 0266.20038)], an alternative proof to the main result of A. Hoare [Q. J. Math., Oxf. II. Ser. 41, No. 161, 45-59 (1990; Zbl 0693.20044)], and it is pointed out how to derive some results on normal subgroups obtained by the reviewer [in Math. Z. 178, 331-341 (1981; Zbl 0451.20047)] and J. A. Bujalance [in Arch. Math. 49, 470-478 (1987; Zbl 0617.20026)].

MSC:

20H15 Other geometric groups, including crystallographic groups
20E07 Subgroup theorems; subgroup growth
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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References:

[1] DOI: 10.1007/BF02782941 · Zbl 0632.20023 · doi:10.1007/BF02782941
[2] Gersten, Essays in Group Theory pp 15– (1987) · doi:10.1007/978-1-4613-9586-7_2
[3] Bujalance, Arch. Math. (Basel) 49 pp 470– (1987) · Zbl 0617.20026 · doi:10.1007/BF01194293
[4] DOI: 10.1007/BF01214870 · Zbl 0451.20047 · doi:10.1007/BF01214870
[5] DOI: 10.1016/0021-8693(89)90195-6 · Zbl 0679.20033 · doi:10.1016/0021-8693(89)90195-6
[6] DOI: 10.1017/S0017089500007175 · Zbl 0684.20022 · doi:10.1017/S0017089500007175
[7] DOI: 10.1017/S0017089500006510 · Zbl 0595.20037 · doi:10.1017/S0017089500006510
[8] Magnus, Combinatorial Group Theory (1976)
[9] DOI: 10.1017/S0305004100052245 · Zbl 0338.20056 · doi:10.1017/S0305004100052245
[10] Lyndon, Combinatorial Group Theory (1977) · doi:10.1007/978-3-642-61896-3
[11] DOI: 10.1002/cpa.3160260515 · Zbl 0266.20038 · doi:10.1002/cpa.3160260515
[12] DOI: 10.1007/BF01109993 · Zbl 0223.20053 · doi:10.1007/BF01109993
[13] DOI: 10.1007/BF01111114 · Zbl 0223.20054 · doi:10.1007/BF01111114
[14] DOI: 10.1007/BF02774082 · Zbl 0584.20026 · doi:10.1007/BF02774082
[15] DOI: 10.1112/plms/s3-57.2.301 · Zbl 0627.20020 · doi:10.1112/plms/s3-57.2.301
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