On aspherical presentations of groups
Author:
Sergei V. Ivanov
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 109114
MSC (1991):
Primary 20F05, 20F06, 20F32; Secondary 57M20
Published electronically:
December 15, 1998
MathSciNet review:
1662323
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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′shanskiĭ, Abelian and central extensions of
aspherical groups, Izv. Vyssh. Uchebn. Zaved. Mat. 11
(1985), 48–60, 85 (Russian). MR 829100
(87m:20095)
 [AC]
J.
J. Andrews and M.
L. Curtis, Free groups and handlebodies,
Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 0173241
(30 #3454), http://dx.doi.org/10.1090/S00029939196501732418
 [B]
S.
D. Brodskiĭ, Equations over groups and groups with one
defining relation, Uspekhi Mat. Nauk 35 (1980),
no. 4(214), 183 (Russian). MR 586195
(82a:20041)
 [GR]
Mauricio
A. Gutiérrez and John
G. Ratcliffe, On the second homotopy group, Quart. J. Math.
Oxford Ser. (2) 32 (1981), no. 125, 45–55. MR 606922
(82g:57003), http://dx.doi.org/10.1093/qmath/32.1.45
 [H1]
James
Howie, Some remarks on a problem of J. H. C. Whitehead,
Topology 22 (1983), no. 4, 475–485. MR 715251
(85g:57003), http://dx.doi.org/10.1016/00409383(83)900381
 [H2]
J. Howie, On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 289 (1985), 281302. 87a:57007
 [H3]
James
Howie, On locally indicable groups, Math. Z.
180 (1982), no. 4, 445–461. MR 667000
(84b:20036), http://dx.doi.org/10.1007/BF01214717
 [Hb1]
Johannes
Huebschmann, Cohomology theory of aspherical groups and of small
cancellation groups, J. Pure Appl. Algebra 14 (1979),
no. 2, 137–143. MR 524183
(80e:20064), http://dx.doi.org/10.1016/00224049(79)900033
 [Hb2]
Johannes
Huebschmann, Aspherical 2complexes and an unsettled problem of J.
H. C. Whitehead, Math. Ann. 258 (1981/82),
no. 1, 17–37. MR 641666
(83e:57004), http://dx.doi.org/10.1007/BF01450344
 [I]
Sergei
V. Ivanov, The free Burnside groups of sufficiently large
exponents, Internat. J. Algebra Comput. 4 (1994),
no. 12, ii+308. MR 1283947
(95h:20051), http://dx.doi.org/10.1142/S0218196794000026
 [IO1]
Sergei
V. Ivanov and Alexander
Yu. Ol′shanskii, Some applications of graded diagrams in
combinatorial group theory, Groups—St. Andrews 1989, Vol. 2,
London Math. Soc. Lecture Note Ser., vol. 160, Cambridge Univ. Press,
Cambridge, 1991, pp. 258–308. MR 1123985
(92j:20022), http://dx.doi.org/10.1017/CBO9780511661846.004
 [IO2]
S.
V. Ivanov and A.
Yu. Ol′shanskiĭ, Hyperbolic groups and their quotients
of bounded exponents, Trans. Amer. Math.
Soc. 348 (1996), no. 6, 2091–2138. MR 1327257
(96m:20057), http://dx.doi.org/10.1090/S0002994796015103
 [K]
Anton
A. Klyachko, A funny property of sphere and equations over
groups, Comm. Algebra 21 (1993), no. 7,
2555–2575. MR 1218513
(94c:20070), http://dx.doi.org/10.1080/00927879308824692
 [Lv]
Frank
Levin, Solutions of equations over
groups, Bull. Amer. Math. Soc. 68 (1962), 603–604. MR 0142643
(26 #212), http://dx.doi.org/10.1090/S000299041962108684
 [Lf]
E.
Luft, On 2dimensional aspherical complexes and a problem of J. H.
C. Whitehead, Math. Proc. Cambridge Philos. Soc. 119
(1996), no. 3, 493–495. MR 1357060
(96h:57003), http://dx.doi.org/10.1017/S0305004100074363
 [Ln]
Roger
C. Lyndon, Cohomology theory of groups with a single defining
relation, Ann. of Math. (2) 52 (1950), 650–665.
MR
0047046 (13,819b)
 [LS]
Roger
C. Lyndon and Paul
E. Schupp, Combinatorial group theory, SpringerVerlag,
Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064
(58 #28182)
 [O1]
A.
Yu. Ol′shanskiĭ, The NovikovAdyan theorem, Mat.
Sb. (N.S.) 118(160) (1982), no. 2, 203–235, 287
(Russian). MR
658789 (83m:20058)
 [O2]
A.
Yu. Ol′shanskiĭ, Groups of bounded period with
subgroups of prime order, Algebra i Logika 21 (1982),
no. 5, 553–618 (Russian). MR 721048
(85g:20052)
 [O3]
A.
Yu. Ol′shanskiĭ, Geometry of defining relations in
groups, Mathematics and its Applications (Soviet Series),
vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated
from the 1989 Russian original by Yu. A. Bakhturin. MR 1191619
(93g:20071)
 [S]
Allan
J. Sieradski, Combinatorial isomorphisms and combinatorial homotopy
equivalences, J. Pure Appl. Algebra 7 (1976),
no. 1, 59–95. MR 0405434
(53 #9227)
 [W]
J.
H. C. Whitehead, On adding relations to homotopy groups, Ann.
of Math. (2) 42 (1941), 409–428. MR 0004123
(2,323c)
 [AO]
 I.S. Ashmanov and A.Yu. Ol'shanskii, On abelian and central extensions of aspherical groups, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1985), 4860. MR 87m:20095
 [AC]
 J.J. Andrews and M.L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192195. 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), 4555. MR 82g:57003
 [H1]
 J. Howie, Some remarks on a problem of J.H.C. Whitehead, Topology 22 (1983), 475485. MR 85g:57003
 [H2]
 J. Howie, On the asphericity of ribbon disc complements, Trans. Amer. Math. Soc. 289 (1985), 281302. 87a:57007
 [H3]
 J. Howie, On locally indicable groups, Math. Z. 180 (1982), 445461. MR 84b:20036
 [Hb1]
 J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra 14 (1979), 137143. MR 80e:20064
 [Hb2]
 J. Huebschmann, Aspherical 2complexes and an unsettled problem of J.H.C. Whitehead, Math. Ann. 258 (1981), 1737. MR 83e:57004
 [I]
 S.V. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comp. 4 (1994), 1308. 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), 258308. 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), 20912138. MR 96m:20057
 [K]
 A. Klyachko, A funny property of sphere and equations over groups, Comm. Algebra 122 (1994), 14751488. MR 94c:20070
 [Lv]
 F. Levin, Solutions of equations over groups, Bull. Amer. Math. Soc. 62 (1962), 603604. MR 26:212
 [Lf]
 E. Luft, On 2dimensional aspherical complexes and a problem of J.H.C. Whitehead, Math. Proc. Cambridge Phil. Soc. 119 (1996), 493495. MR 96h:57003
 [Ln]
 R.C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. Math. 52 (1950), 650655. MR 13:819b
 [LS]
 R.C. Lyndon and P.E. Schupp, Combinatorial group theory, SpringerVerlag, 1977. MR 58:28182
 [O1]
 A.Yu. Ol'shanskii, On the NovikovAdian theorem, Mat. Sbornik 118 (1982), 203235. MR 83m:20058
 [O2]
 A.Yu. Ol'shanskii, Groups of bounded period with subgroups of prime order, Algebra i Logika 21 (1982), 553618. 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), 5965. MR 53:9227
 [W]
 J.H.C. Whitehead, On adding relations to homotopy groups, Ann. Math. 42 (1941), 409428. MR 2:323c
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (1991):
20F05,
20F06,
20F32,
57M20
Retrieve articles in all journals
with MSC (1991):
20F05,
20F06,
20F32,
57M20
Additional Information
Sergei V. Ivanov
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 West Green Street, Urbana, IL 61801
Email:
ivanov@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S1079676298000523
PII:
S 10796762(98)000523
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 9501056
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 1998 American Mathematical Society
