zbMATH — the first resource for mathematics

A generalized weight test with applications to tree presentations. (Ein verallgemeinerter Gewichtstest mit Anwendungen auf Baumpräsentationen.) (German) Zbl 0761.57003
We introduce the “cycle test”, which is a generalization of Gersten’s “weight test”, to study diagrammatic reducibility of standard 2- complexes. Diagrammatic reducibility (DR) is a stronger combinatorial form of asphericity for 2-complexes. A homogeneous version of the cycle test leads to conditions that generalize small cancellation theory in its application as a test for DR; a fact which also has been observed by Gersten. Aside from that, the cycle test has many applications in a nonhomogeneous way, of which we present several examples. In particular, some of these examples are related to “labelled oriented tree” presentations whose 2-complexes are spines of ribbon disk complements.
Reviewer: G.Huck

57M20 Two-dimensional complexes (manifolds) (MSC2010)
20F06 Cancellation theory of groups; application of van Kampen diagrams
57M15 Relations of low-dimensional topology with graph theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
Full Text: DOI EuDML
[1] Collins, D.J., Huebschmann, J.: Spherical diagrams and identities among relations. Math. Ann.261, 155–183 (1982) · Zbl 0477.20019 · doi:10.1007/BF01456216
[2] Fenn, R.: Techniques of geometric topology. Lond. Math. Proc. Lect. Note Ser.57 (1978) · Zbl 0517.57001
[3] Gersten, S.: Reducible diagrams and equations over groups. In: Gersten, S. (ed.) Essays in group theory. Publ., Math. Sci. Res. Inst., vol. 8, pp. 15–73, Berlin Heidelberg New York: Springer 1987
[4] Gersten, S.: The isoperimetric inequality and the word problem (Unpublished, 1988)
[5] Howie, J.: Some remarks on a problem of J.H.C.. Whitehead. Topology22, 475–485 (1983) · Zbl 0524.57002 · doi:10.1016/0040-9383(83)90038-1
[6] Howie, J.: Spherical diagrams and equations over groups. Math. Proc. Camb. Philos. Soc.96, 255–268 (1984) · Zbl 0542.20013 · doi:10.1017/S0305004100062150
[7] Howie, J.: On the asphericity of ribbon disk complements. Trans. Am. Math. Soc., I. Ser.289, 281–302 (1985) · Zbl 0572.57001 · doi:10.1090/S0002-9947-1985-0779064-8
[8] Juhasz, A.: Small cancellation theory with a weakened small cancellation hypothesis. 1. The basic theory. Isr. J. Math.55 (1), 65–93 (1986) · Zbl 0603.20028 · doi:10.1007/BF02772696
[9] Juhasz, A.: Small cancellation theory with a unified small cancellation condition. J. Lond. Math. Soc., II. Ser.40, 57–80 (1989) · Zbl 0647.20030 · doi:10.1112/jlms/s2-40.1.57
[10] Lyndon, R.: On Dehn’s algorithm. Math. Ann.166, 208–228 (1966) · Zbl 0138.25702 · doi:10.1007/BF01361168
[11] Lyndon, R., Schupp, P.: Combinatorial group theory. Berlin Heidelberg New York: Springer 1977 · Zbl 0368.20023
[12] Metzler, W.: Über den Homotopietyp zweidimensionaler CW-Komplexe und Elementartransformationen bei Darstellungen von Gruppen durch Erzeugende und definierende Relationen. J. Reine Angew. Math.285, 7–23 (1976) · Zbl 0325.57003 · doi:10.1515/crll.1976.285.7
[13] Pride, S.: The diagrammatic asphericity of groups given by presentations in which each defining relator involves exactly two types of generators. Arch. Math.50, 570–574 (1988) · Zbl 0655.20023 · doi:10.1007/BF01193628
[14] Reshetnyak, Y.G.: On a special kind of mapping of a cone onto a polyhedral disk. Math. Sb. 53 (95), 39–52 (1961); engl. Uebersetzung: Berkeley: J. Stallings UC
[15] Rosebrock, S.: A reduced spherical diagram into a ribbon-disk complement and related examples. In: Latiolais, P. (ed.) Topology and Combinatorial Groups Theory. (Lect. Notes Math., vol. 1440, pp. 175–185) Berlin Heidelberg New York: Springer 1990 · Zbl 0714.57001
[16] Sieradski, A.: A coloring test for asphericity. Q. J. Math., Oxf., II. Ser.34, 97–106 (1983) · Zbl 0522.57003 · doi:10.1093/qmath/34.1.97
[17] Van, Do Long: On the word and conjugacy problems for some classes of finitely presented groups. Dokl. Acad. Nauk SSSR241, 5 (1978) · Zbl 0424.20032
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.