Abstract
Diagrammatic reducibility is related to the solution of equations over groups. Sufficient conditions for the reducibility of all spherical diagrams are given, unifying and generalizing work of Adian, Remmers, Lyndon, and Sieradski. Hyperbolic 2-complexes are defined and the word problem is solved for their fundamental groups.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
S.J. Adian, Defining relations and algorithmic problems for groups and semigroups, Proc. Steklov Inst. Math. 85 (1966).
I.M. Chiswell, D.J. Collins, and J. Huebschmann, Aspherical group presentations, Math Z. 178 (1981), 1–36.
D.J. Collins and J. Huebschmann, Spherical diagrams and identities among relations, Math. Ann. 261 (1983), 155–183.
E. Formanek, Conjugate separability in polycyclic groups, J. Algebra 42 (1976), 1–10.
S.M. Gersten, Conservative groups, indicability and a conjecture of Howie, J. Pure Appl. Alg. 29 (1983), 59–74.
S.M. Gersten, Nonsingular equations of small weight over groups, to appear in Proceedings of Alta Conference on Combinatorial Group Theory, Annals of Math Studies, Princeton.
S.M. Gersten, Products of conjugacy classes in a free group; a counterexample, to appear in Math Z.
M. Gerstenhaber and O.S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. USA 48 (1962), 1531–1533.
J. Howie, How to generalize one — relator group theory, to appear in Proceedings of Alta Conference on Combinatorial Group Theory, Annals of Math Studies, Princeton.
J. Howie, On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 289 (1985), 281–302.
J. Howie, On pairs of 2-complexes and systems of equations over groups, Crelle 324 (1981), 165–174.
J. Howie, The p-adic typology on a free group: a counterexample, Math. Z. 187 (1984), 25–27.
J. Howie, The solution of length three equations over groups, Proc. Edinburgh Math. Soc. 26:2 (1981), 165–174.
F. Levin, Solutions of equations over groups, Bull. Amer. Math. Soc. 68 (1982), 603–604.
R.C. Lyndon, Problems in combinatorial group theory, to appear in Proceedings of Alta Conference on Combinatorial Group Theory, Annals of Math Studies, Princeton.
R.C. Lyndon and P.E. Schupp, Combinatorial group theory, Ergebnisse der Math. 89 (1977), Springer-Verlag.
Ch.D. Papakyriakopoulos, Bull. Soc. Math. Grèce 22 (1943), 1–154.
V.N. Remeslennikov, Finite approximability of groups with respect to conjugacy, Sibirsk. Mat. Zh. 12 (1971), 1085–1099.
J.H. Remmers, On the geometry of semigroup presentations, Adv. in Math. 36 (1980), 283–296.
A. Sieradski, A coloring test for asphericity, Quart. J. Math Oxford (2) 34 (1983), 97–106.
J.R. Stallings, A graph-theoretic lemma and group embeddings, to appear in Proc. of Alta Conference on Combinatorial Group Theory, Annals of Math Studies, Princeton.
J.R. Stallings, Surfaces in three-manifolds and non singular equations in groups, Math Z. 184 (1983), 1–17.
J.H.C. Whitehead, On adding relations to homotopy groups, Ann. of Math. 42 (1941), 409–428.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Gersten, S.M. (1987). Reducible Diagrams and Equations Over Groups. In: Gersten, S.M. (eds) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9586-7_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9586-7_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9588-1
Online ISBN: 978-1-4613-9586-7
eBook Packages: Springer Book Archive