On aspherical presentations of groups

Author:
Sergei V. Ivanov

Journal:
Electron. Res. Announc. Amer. Math. Soc. **4** (1998), 109-114

MSC (1991):
Primary 20F05, 20F06, 20F32; Secondary 57M20

DOI:
https://doi.org/10.1090/S1079-6762-98-00052-3

Published electronically:
December 15, 1998

MathSciNet review:
1662323

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Abstract: The Whitehead asphericity conjecture claims that if is an aspherical group presentation, then for every the subpresentation is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction of almost aspherical presentations. It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture and holds for standard presentations of free Burnside groups of large odd exponent, Tarski monsters and some others. Next, it is proven that if the Whitehead conjecture is false, then there is an aspherical presentation of the trivial group , where the alphabet is finite or countably infinite and , such that its subpresentation is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite and ), then there is a finite aspherical presentation , , such that for every the subpresentation is aspherical and the subpresentation of aspherical is not aspherical. Now suppose a group presentation is aspherical, , is a word in the alphabet with nonzero sum of exponents on , and the group naturally embeds in . It is conjectured that the presentation is aspherical if and only if is torsion free. It is proven that if this conjecture is false and is a counterexample, then the integral group ring of the torsion free group will contain zero divisors. Some special cases where this conjecture holds are also indicated.

**[AO]**I.S. Ashmanov and A.Yu. Ol'shanskii,*On abelian and central extensions of aspherical groups*, Izv. Vyssh. Uchebn. Zaved. Mat.**11**(1985), 48-60. MR**87m:20095****[AC]**J.J. Andrews and M.L. Curtis,*Free groups and handlebodies*, Proc. Amer. Math. Soc.**16**(1965), 192-195. MR**30:3454****[B]**S.D. Brodskii,*Equations over groups and groups with a single defining relation*, Uspekhi Mat. Nauk**35**(1980), 183. MR**82a:20041****[GR]**M. Gutierrez and J.G. Ratcliffe,*On the second homotopy group*, Quart. J. Math. Oxford**32**(1981), 45-55. MR**82g:57003****[H1]**J. Howie,*Some remarks on a problem of J.H.C. Whitehead*, Topology**22**(1983), 475-485. MR**85g:57003****[H2]**J. Howie,*On the asphericity of ribbon disc complements*, Trans. Amer. Math. Soc.**289**(1985), 281-302. 87a:57007**[H3]**J. Howie,*On locally indicable groups*, Math. Z.**180**(1982), 445-461. MR**84b:20036****[Hb1]**J. Huebschmann,*Cohomology theory of aspherical groups and of small cancellation groups*, J. Pure Appl. Algebra**14**(1979), 137-143. MR**80e:20064****[Hb2]**J. Huebschmann,*Aspherical 2-complexes and an unsettled problem of J.H.C. Whitehead*, Math. Ann.**258**(1981), 17-37. MR**83e:57004****[I]**S.V. Ivanov,*The free Burnside groups of sufficiently large exponents*, Internat. J. Algebra Comp.**4**(1994), 1-308. MR**95h:20051****[IO1]**S.V. Ivanov and A.Yu. Ol'shanskii,*Some applications of graded diagrams in combinatorial group theory*, London Math. Soc. Lecture Note Ser.**160**(1991), 258-308. MR**92j:20022****[IO2]**S.V. Ivanov and A.Yu. Ol'shanskii,*Hyperbolic groups and their quotients of bounded exponents*, Trans. of the Amer. Math. Soc**348**(1996), 2091-2138. MR**96m:20057****[K]**A. Klyachko,*A funny property of sphere and equations over groups*, Comm. Algebra**122**(1994), 1475-1488. MR**94c:20070****[Lv]**F. Levin,*Solutions of equations over groups*, Bull. Amer. Math. Soc.**62**(1962), 603-604. MR**26:212****[Lf]**E. Luft,*On 2-dimensional aspherical complexes and a problem of J.H.C. Whitehead*, Math. Proc. Cambridge Phil. Soc.**119**(1996), 493-495. MR**96h:57003****[Ln]**R.C. Lyndon,*Cohomology theory of groups with a single defining relation*, Ann. Math.**52**(1950), 650-655. MR**13:819b****[LS]**R.C. Lyndon and P.E. Schupp,*Combinatorial group theory*, Springer-Verlag, 1977. MR**58:28182****[O1]**A.Yu. Ol'shanskii,*On the Novikov-Adian theorem*, Mat. Sbornik**118**(1982), 203-235. MR**83m:20058****[O2]**A.Yu. Ol'shanskii,*Groups of bounded period with subgroups of prime order*, Algebra i Logika**21**(1982), 553-618. MR**85g:20052****[O3]**A.Yu. Ol'shanskii,*Geometry of defining relations in groups*, English translation in Math. and Its Applications (Soviet series), 70, Kluwer Acad. Publishers, 1991 (1989). MR**93g:20071****[S]**A.J. Sieradski,*Combinatorial isomorphisms and combinatorial homotopy equivalences*, J. Pure Appl. Algebra**7**(1976), 59-65. MR**53:9227****[W]**J.H.C. Whitehead,*On adding relations to homotopy groups*, Ann. Math.**42**(1941), 409-428. MR**2:323c**

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Additional Information

**Sergei V. Ivanov**

Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801

Email:
ivanov@math.uiuc.edu

DOI:
https://doi.org/10.1090/S1079-6762-98-00052-3

Received by editor(s):
April 13, 1998

Published electronically:
December 15, 1998

Additional Notes:
Supported in part by an Alfred P. Sloan Research Fellowship and NSF grant DMS 95-01056

Communicated by:
Efim Zelmanov

Article copyright:
© Copyright 1998
American Mathematical Society