Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
Aspherical Relative Presentations all Over Again
Published online by Cambridge University Press: 15 April 2019
Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
Summary
The concept of asphericity for relative group presentations was introduced twenty-five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged. In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four.
- Type
- Chapter
- Information
- Groups St Andrews 2017 in Birmingham , pp. 169 - 199Publisher: Cambridge University PressPrint publication year: 2019
References
Ahmad, Abdul Ghafur Bin. The application of pictures to decision problems and relative presentations. PhD thesis, University of Glasgow, 1995.Google Scholar
Ahmad, Abd Ghafur Bin, Al-Mulla, Muna A., and Edjvet, Martin. Asphericity of length four relative group presentations. J. Algebra Appl., 16(4):1750076, 2017. (27 pages).Google Scholar
Aldwaik, Suzana and Edjvet, Martin. On the asphericity of a family of positive relative group presentations. Proc. Edinb. Math. Soc., 60:545–564, 2017.CrossRefGoogle Scholar
Aldwaik, Suzana, Edjvet, Martin, and Juhasz, Arye. Asphericity of positive free product length 4 relative group presentations. Forum Math., to appear.Google Scholar
Anshel, Iris Lee. On two relator groups. In Latiolais, Paul, editor, Topology and Combinatorial Group Theory, Proceedings of the Fall Foliage Topology Seminars held in New Hampshire 1985-1988, volume 1440 of Lecture Notes Math., pages 1–21. Springer-Verlag Berlin Heidelberg, 1990.Google Scholar
Baik, Y.G., William, A. Bogley, and Pride, Stephen J.. On the asphericity of length four relative group presentations. Int. J. Algebra Comput., 7(3):277–312, 1997.CrossRefGoogle Scholar
Bardakov, V. G. and Vesnin, A.Yu.. A generalization of Fibonacci groups. Algebra Logika, 42(2):131–160, 2003. Translation in Algebra Logic 42(2):73–91 (2003).Google Scholar
Besche, A. U., Eick, B., and O’Brien, E.A.. The SmallGroups Library – a GAP package, 2002.Google Scholar
Bogley, William A.. An identity theorem for multi-relator groups. Math. Proc. Camb. Philos. Soc., 109(2):313–321, 1991.CrossRefGoogle Scholar
Bogley, William A.. On shift dynamics for cyclically presented groups. J. Algebra, 418:154–173, 2014.CrossRefGoogle Scholar
Bogley, William A. and Parker, Forrest W.. Cyclically presented groups with length four positive relators. J. Group Theory, 21:911–948, 2018.CrossRefGoogle Scholar
Bogley, William A. and Pride, Stephen J.. Aspherical relative presentations. Proc. Edin. Math. Soc., 35(1):1–39, 1992.CrossRefGoogle Scholar
Bogley, William A. and Pride, Steve J.. Calculating generators of π2. In Hog-Angeloni, C., Metzler, W., and Sieradski, A.J., editors, Two-dimensional homotopy and combinatorial group theory, volume 197 of London Math. Soc. Lecture Note Ser., pages 157–188. Cambridge: Cambridge University Press, 1993.Google Scholar
Bogley, William A. and Williams, Gerald. Efficient finite groups arising in the study of relative asphericity. Math. Z., 284(1-2):507–535, 2016.CrossRefGoogle Scholar
Bogley, William A. and Williams, Gerald. Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups. J. Algebra, 480:266–297, 2017.CrossRefGoogle Scholar
Brown, Kenneth S.. Cohomology of groups, volume 87 of Graduate Text Math. Springer New York, 1982.CrossRefGoogle Scholar
Brown, Kenneth S.. Lectures on the cohomology of groups. In Cohomology of groups and algebraic K-theory. Selected papers of the international summer school on cohomology of groups and algebraic K-theory, Hangzhou, China, July 1–3, 2007, pages 131–166. Somerville, MA: International Press; Beijing: Higher Education Press, 2010.Google Scholar
Cavicchioli, A., Hegenbarth, F., and Kim, A. C.. A geometric study of Sieradski groups. Algebra Colloq., 5(2):203–217, 1998.Google Scholar
Cavicchioli, Alberto, E. A. O’Brien, and Spaggiari, Fulvia. On some questions about a family of cyclically presented groups. J. Algebra, 320(11):4063–4072, 2008.CrossRefGoogle Scholar
Chalk, Christopher P.. Fibonacci groups with aspherical presentations. Commun. Algebra, 26(5):1511–1546, 1998.CrossRefGoogle Scholar
Chiswell, Ian M., Donald, J. Collins, and Huebschmann, Johannes. Aspherical group presentations. Math. Z., 178:1–36, 1981.CrossRefGoogle Scholar
Corson, J. M. and Trace, B.. Diagrammatically reducible complexes and Haken manifolds. J. Aust. Math. Soc., Ser. A, 69(1):116–126, 2000.Google Scholar
Davidson, Peter J.. On the asphericity of a family of relative group presentations. Int. J. Algebra Comput., 19(2):159–189, 2009.CrossRefGoogle Scholar
Dyer, Eldon and Vasquez, A. T.. Some small aspherical spaces. J. Aust. Math. Soc., 16:332–352, 1973.CrossRefGoogle Scholar
Edjvet, Martin. Equations over groups and a theorem of Higman, Neumann and Neumann. Proc. Lond. Math. Soc. (3), 63(3):563–589, 1991.Google Scholar
Edjvet, Martin. Solutions of certain sets of equations over groups. In Groups St Andrews 1989, Volume 1, number 159 in Lecture Note Ser., pages 105–123. London Math. Soc., Cambridge University Press, 1991.Google Scholar
Edjvet, Martin. On the asphericity of one-relator relative presentations. Proc. R. Soc. Edinb., Sect. A, 124(4):713–728, 1994.CrossRefGoogle Scholar
Edjvet, Martin and Juhasz, Arye. The infinite Fibonacci groups and relative asphericity. Trans. Lond. Math. Soc., 4(1):148–218, 2017.Google Scholar
Forester, Max and Rourke, Colin. A fixed-point theorem and relative asphericity. Enseign. Math. (2), 51(3-4):231–237, 2005.Google Scholar
Freedman, Michael H.. Remarks on the solution of first degree equations in groups. In Millet, K. C., editor, Algebraic and Geometric Topology Proceedings of a Symposium held at Santa Barbara in honor of Raymond L. Wilder, July 25–29, 1977, volume 664 of Lect. Notes Math., pages 87–93. Springer, 1978.Google Scholar
Gersten, S. M.. On certain equations over torsion free groups.Google Scholar
Gersten, S. M.. Reducible diagrams and equations over groups. In Gersten, S. M., editor, Essays in Group Theory, volume 8 of Publ. Math. Sci. Res. Inst., pages 15–73. Springer, New York, 1987.CrossRefGoogle Scholar
Gerstenhaber, M. and Rothaus, O. S.. The solution of sets of equations in groups. Proc. Natl. Acad. Sci. USA, 48:1531–1533, 1962.CrossRefGoogle ScholarPubMed
Gilbert, N. D. and Howie, James. LOG groups and cyclically presented groups. J. Algebra, 174(1):118–131, 1995.CrossRefGoogle Scholar
Helling, H., Kim, A. C., and Mennicke, J. L.. A geometric study of Fibonacci groups. J. Lie Theory, 8(1):1–23, 1998.Google Scholar
Howie, James. On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math., 324:165–174, 1981.Google Scholar
Howie, James. The solution of length three equations over groups. Proc. Edinb. Math. Soc., II. Ser., 26:89–96, 1983.Google Scholar
Howie, James. How to generalize one-relator group theory. In Gersten, S. M. and Stallings, John R., editors, Combinatorial group theory and topology, volume 111 of Annals of Mathematics Studies, pages 53–78. Princeton University Press, 1987.Google Scholar
Howie, James. The quotient of a free product of groups by a single high-powered relator. I: Pictures. Fifth and higher powers. Proc. Lond. Math. Soc. (3), 59(3):507–540, 1989.Google Scholar
Howie, James. Nonsingular systems of two length three equations over a group. Math. Proc. Camb. Philos. Soc., 110(1):11–24, 1991.CrossRefGoogle Scholar
Howie, James and Metaftsis, V.. On the asphericity of length five relative group presentations. Proc. Lond. Math. Soc. (3), 82(1):173–194, 2001.CrossRefGoogle Scholar
Howie, James and Hans Rudolf Schneebeli. Permutation modules and projective resolutions. Comment. Math. Helv., 56:447–464, 1981.CrossRefGoogle Scholar
Howie, James and Williams, Gerald. Fibonacci type presentations and 3-manifolds. Topology Appl., 215:24–34, 2017.CrossRefGoogle Scholar
Huebschmann, Johannes. Cohomology theory of aspherical groups and of small cancellation groups. J. Pure Appl. Algebra, 14:137–143, 1979.CrossRefGoogle Scholar
Ivanov, S. V.. An asphericity conjecture and Kaplansky problem on zero divisors. J. Algebra, 216(1):13–19, 1999.CrossRefGoogle Scholar
Johnson, D. L.. Topics in the Theory of Group Presentations, volume 42 of London Math. Soc. Lecture Note Ser. Cambridge: Cambridge University Press, 1980.CrossRefGoogle Scholar
Kaplansky, Irving. Fields and rings. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 2nd edition, 1972.Google Scholar
Kim, Seong Kun. On the asphericity of certain relative presentations over torsion-free groups. Int. J. Algebra Comput., 18(6):979–987, 2008.CrossRefGoogle Scholar
Kim, Seong Kun. On the asphericity of length-6 relative presentations with torsion-free coefficients. Proc. Edinb. Math. Soc., II. Ser., 51(1):201–214, 2008.Google Scholar
Klyachko, Anton A.. A funny property of sphere and equations over groups. Comm. Algebra, 21(7):2555–2575, 1993.CrossRefGoogle Scholar
Klyachko, Anton A.. The Kervaire-Laudenbach conjecture and presentations of simple groups. Algebra Logika, 44(4):399–437, 2005.CrossRefGoogle Scholar
Leary, Ian J.. Asphericity and zero divisors in group algebras. J. Algebra, 227(1):362–364, 2000.CrossRefGoogle Scholar
Lyndon, Roger C.. Cohomology theory of groups with a single defining relation. Ann. Math. (2), 52:650–665, 1950.CrossRefGoogle Scholar
Lyndon, Roger C. and Schupp, Paul E.. Combinatorial group theory. Berlin: Springer, reprint of the 1977 edition, 2001.CrossRefGoogle Scholar
Magnus, Wilhelm, Karrass, Abraham, and Solitar, Donald. Combinatorial group theory. Presentations of groups in terms of generators and relations. Dover Books on Advanced Mathematics. Dover Publications, 2nd edition, 1976.Google Scholar
McDermott, Kirk M.. Topological and Dynamical Properties of Cyclically Presented Groups. PhD thesis, Oregon State University, 2017.Google Scholar
Metaftsis, V.. On the asphericity of relative group presentations of arbitrary length. Int. J. Algebra Comput., 13(3):323–339, 2003.CrossRefGoogle Scholar
Montgomery, M. S.. Left and right inverses in group algebras. Bull. Am. Math. Soc., 75:539–540, 1969.CrossRefGoogle Scholar
Odoni, R. W. K.. Some Diophantine problems arising from the theory of cyclicallypresented groups. Glasg. Math. J., 41(2):157–165, 1999.CrossRefGoogle Scholar
Ol’shanskij, A. Yu. On a geometric method in the combinatorial group theory. In Proc. Int. Congr. Math., Warszawa 1983, volume 1, pages 415–424. Warszawa: PWNPolish Scientific Publishers, 1984.Google Scholar
Ol’shanskij, A. Yu. Geometry of defining relations in groups. Dordrecht etc.: Kluwer Academic Publishers, 1991.CrossRefGoogle Scholar
Prishchepov, M. I.. Asphericity, atoricity and symmetrically presented groups. Comm. Algebra, 23(13):5095–5117, 1995.CrossRefGoogle Scholar
Ratcliffe, John G.. Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, C). J. Pure Appl. Algebra, 44:303–314, 1987.CrossRefGoogle Scholar
Sieradski, Allan J.. A coloring test for asphericity. Q. J. Math., Oxf. II. Ser., 34:97–106, 1983.CrossRefGoogle Scholar
Sieradski, Allan J.. Combinatorial squashings, 3-manifolds, and the third homology of groups. Invent. Math., 84:121–139, 1986.CrossRefGoogle Scholar
Sieradski, Allan J.. Algebraic topology for two dimensional complexes. In Hog-Angeloni, C., Metzler, W., and Sieradski, A.J., editors, Two-dimensional Homotopy and Combinatorial Group Theory, number 197 in London Math. Soc. Lecture Note Ser., pages 51–96. Cambridge: Cambridge University Press, 1993.Google Scholar
Wall, C. T. C.. Rational Euler characteristics. Proc. Camb. Philos. Soc., 57:182–184, 1961.CrossRefGoogle Scholar
Whitehead, J. H. C.. Combinatorial homotopy. II. Bull. Am. Math. Soc., 55:453–496, 1949.Google Scholar
Williams, Gerald. The aspherical Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups. J. Group Theory, 12(1):139–149, 2009.CrossRefGoogle Scholar
Williams, Gerald. Unimodular integer circulants associated with trinomials. Int. J. Number Theory, 6(4):869–876, 2010.CrossRefGoogle Scholar
Williams, Gerald. Groups of Fibonacci type revisited. Int. J. Algebra Comput., 22(8), 2012.CrossRefGoogle Scholar
- 3
- Cited by