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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

##### MSC:
 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
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##### References:
 [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15–73, Berlin Heidelberg New York: Springer 1987 [4] Gersten, S.: The isoperimetric inequality and the word problem (Unpublished, 1988) [5] Howie, J.: Some remarks on a problem of J.H.C..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
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