Reducible diagrams and equations over groups.

*(English)*Zbl 0644.20024
Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 15-73 (1987).

[For the entire collection see Zbl 0626.00014.]

Let X be a combinatorial 2-complex, let S be a tesselated sphere, and let \(f: S\to X\) be a combinatorial map. In the paper under review such a pair (S,f) is called a spherical diagram; it is said to be reducible, if it has the following:

Property. S has two distinct faces F and F’ having an edge e in common, and there is an orientation reversing homeomorphism \(g: F\to F'\) fixing \(F\cap F'\) pointwise so that \(f| F'=f| F\circ g.\)

A combinatorial 2-complex is said to be diagrammatically reducible if every spherical diagram is reducible. In the paper, these concepts are applied to the study of the so-called Kervaire-Laudenbach conjecture that a non-singular system of equations over a group G has a solution in an overgroup \(\bar G,\) see e.g. [J. Howie, J. Reine Angew. Math. 324, 165-174 (1981; Zbl 0447.20032)], and of a conjecture of J. Stallings [Math. Z. 184, 1-17 (1983; Zbl 0496.57006)]. The latter conjecture would imply the former, but a counterexample found by the author [to appear] shows the former is false in general. A statement equivalent to that of the Stallings conjecture is what is called the “reciprocity law”. The latter is the basic objective of study in the paper under review. It is proved (Theorem 3.3) that a diagrammatically reducible 2-complex X satisfies the reciprocity law. Furthermore, A. Sieradski’s “coloring test” [Q. J. Math., Oxf. II. Ser. 34, 97-106 (1983; Zbl 0522.57003)] is extended to what is called the weight test in Section 4. A 2-complex which passes the weight test is shown to be diagrammatically reducible. This is also related with results of Lyndon, S. I. Adyan [Tr. Mat. Inst. Steklova 85 (1966; Zbl 0204.01702)], and J. Remmers [Adv. Math. 36, 283-296 (1980; Zbl 0438.20041)]. Finally, it is shown that staggered 2-complexes satisfy the reciprocity law, and combinatorial invariance of the reciprocity law, diagrammatic reducibility, and the weight test are examined. In an appendix a notion of hyperbolic 2-complexes is proposed, and the word problem for the corresponding presentation of fundamental group is solved.

{Reviewer’s remarks: The notion of a spherical diagram in [D. Collins and the reviewer, Math. Ann. 261, 155-183 (1982; Zbl 0477.20019)] differs from the one in the paper under review in that in [loc. cit.] more general spherical diagrams (involving among others several “tailed spheres” joined at a common basepoint) come into play. Furthermore, the present notion of reducibility differs from the one in [loc. cit.]. For these matters, see our review about the author’s paper [Trans. Am. Math. Soc. 303, 689-706 (1987; reviewed above)].}

Let X be a combinatorial 2-complex, let S be a tesselated sphere, and let \(f: S\to X\) be a combinatorial map. In the paper under review such a pair (S,f) is called a spherical diagram; it is said to be reducible, if it has the following:

Property. S has two distinct faces F and F’ having an edge e in common, and there is an orientation reversing homeomorphism \(g: F\to F'\) fixing \(F\cap F'\) pointwise so that \(f| F'=f| F\circ g.\)

A combinatorial 2-complex is said to be diagrammatically reducible if every spherical diagram is reducible. In the paper, these concepts are applied to the study of the so-called Kervaire-Laudenbach conjecture that a non-singular system of equations over a group G has a solution in an overgroup \(\bar G,\) see e.g. [J. Howie, J. Reine Angew. Math. 324, 165-174 (1981; Zbl 0447.20032)], and of a conjecture of J. Stallings [Math. Z. 184, 1-17 (1983; Zbl 0496.57006)]. The latter conjecture would imply the former, but a counterexample found by the author [to appear] shows the former is false in general. A statement equivalent to that of the Stallings conjecture is what is called the “reciprocity law”. The latter is the basic objective of study in the paper under review. It is proved (Theorem 3.3) that a diagrammatically reducible 2-complex X satisfies the reciprocity law. Furthermore, A. Sieradski’s “coloring test” [Q. J. Math., Oxf. II. Ser. 34, 97-106 (1983; Zbl 0522.57003)] is extended to what is called the weight test in Section 4. A 2-complex which passes the weight test is shown to be diagrammatically reducible. This is also related with results of Lyndon, S. I. Adyan [Tr. Mat. Inst. Steklova 85 (1966; Zbl 0204.01702)], and J. Remmers [Adv. Math. 36, 283-296 (1980; Zbl 0438.20041)]. Finally, it is shown that staggered 2-complexes satisfy the reciprocity law, and combinatorial invariance of the reciprocity law, diagrammatic reducibility, and the weight test are examined. In an appendix a notion of hyperbolic 2-complexes is proposed, and the word problem for the corresponding presentation of fundamental group is solved.

{Reviewer’s remarks: The notion of a spherical diagram in [D. Collins and the reviewer, Math. Ann. 261, 155-183 (1982; Zbl 0477.20019)] differs from the one in the paper under review in that in [loc. cit.] more general spherical diagrams (involving among others several “tailed spheres” joined at a common basepoint) come into play. Furthermore, the present notion of reducibility differs from the one in [loc. cit.]. For these matters, see our review about the author’s paper [Trans. Am. Math. Soc. 303, 689-706 (1987; reviewed above)].}

Reviewer: J.Huebschmann

##### MSC:

20F05 | Generators, relations, and presentations of groups |

57M20 | Two-dimensional complexes (manifolds) (MSC2010) |