Nonpositive immersions, sectional curvature, and subgroup properties
Author:
Daniel T. Wise
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 1-9
MSC (2000):
Primary 20F05, 20F67, 57M07, 57M20
DOI:
https://doi.org/10.1090/S1079-6762-03-00105-7
Published electronically:
January 10, 2003
MathSciNet review:
1988866
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Additional Information
Abstract: This announcement describes research concerning local quasiconvexity, coherence, compact cores, and local indicability for fundamental groups of certain $2$-complexes.
- W. Ballmann and S. Buyalo, Nonpositively curved metrics on $2$-polyhedra, Math. Z. 222 (1996), no. 1, 97–134. MR 1388005, DOI https://doi.org/10.1007/PL00004529
- Gilbert Baumslag, Some problems on one-relator groups, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Springer, Berlin, 1974, pp. 75–81. Lecture Notes in Math., Vol. 372. MR 0364463
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
- S. D. Brodskiĭ, Equations over groups and groups with one defining relation, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 183 (Russian). MR 586195
- S. D. Brodskiĭ, Equations over groups, and groups with one defining relation, Sibirsk. Mat. Zh. 25 (1984), no. 2, 84–103 (Russian). MR 741011
- R. G. Burns and V. W. D. Hale, A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972), 441–445. MR 310046, DOI https://doi.org/10.4153/CMB-1972-080-3
- J. M. Corson and B. Trace, Diagrammatically reducible complexes and Haken manifolds, J. Austral. Math. Soc. Ser. A 69 (2000), no. 1, 116–126. MR 1767395
- Warren Dicks, Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture, Invent. Math. 117 (1994), no. 3, 373–389. MR 1283723, DOI https://doi.org/10.1007/BF01232249
- D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), no. 1, 67–80. MR 918457
- Mark Feighn and Michael Handel, Mapping tori of free group automorphisms are coherent, Ann. of Math. (2) 149 (1999), no. 3, 1061–1077. MR 1709311, DOI https://doi.org/10.2307/121081
- J. Fischer, A. Karrass, and D. Solitar, On one-relator groups having elements of finite order, Proc. Amer. Math. Soc. 33 (1972), 297–301. MR 311780, DOI https://doi.org/10.1090/S0002-9939-1972-0311780-0
- S. M. Gersten, Reducible diagrams and equations over groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 15–73. MR 919828, DOI https://doi.org/10.1007/978-1-4613-9586-7_2
- James Howie, On pairs of $2$-complexes and systems of equations over groups, J. Reine Angew. Math. 324 (1981), 165–174. MR 614523, DOI https://doi.org/10.1515/crll.1981.324.165
- James Howie, On locally indicable groups, Math. Z. 180 (1982), no. 4, 445–461. MR 667000, DOI https://doi.org/10.1007/BF01214717
- James Howie, How to generalize one-relator group theory, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 53–78. MR 895609
- Johannes Huebschmann, Aspherical $2$-complexes and an unsettled problem of J. H. C. Whitehead, Math. Ann. 258 (1981/82), no. 1, 17–37. MR 641666, DOI https://doi.org/10.1007/BF01450344
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064
Magnus30 Wilhelm Magnus. Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz). J. Reine Angew. Math., 163:141–165, 1930.
Magnus32 Wilhelm Magnus. Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann., 106:295–307, 1932.
McCammondWiseCoherence Jonathan P. McCammond and Daniel T. Wise. Coherence, local quasiconvexity and the perimeter of $2$-complexes. Preprint, 1999.
McCammondWiseFanLadder Jonathan P. McCammond and Daniel T. Wise. Fans and ladders in small cancellation theory. Proc. London Math. Soc. (3), 84(3):599–644, 2002.
- Walter D. Neumann, On intersections of finitely generated subgroups of free groups, Groups—Canberra 1989, Lecture Notes in Math., vol. 1456, Springer, Berlin, 1990, pp. 161–170. MR 1092229, DOI https://doi.org/10.1007/BFb0100737
Papakyriakopoulos57 C. D. Papakyriakopoulos. On Dehn’s lemma and the asphericity of knots. Ann. of Math. (2), 66:1–26, 1957.
- Frédéric Paulin, Constructions of hyperbolic groups via hyperbolizations of polyhedra, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 313–372. MR 1170371
- Stephen J. Pride, Star-complexes, and the dependence problems for hyperbolic complexes, Glasgow Math. J. 30 (1988), no. 2, 155–170. MR 942986, DOI https://doi.org/10.1017/S0017089500007175
- E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45–47. MR 642423, DOI https://doi.org/10.1112/blms/14.1.45
- G. P. Scott, Finitely generated $3$-manifold groups are finitely presented, J. London Math. Soc. (2) 6 (1973), 437–440. MR 380763, DOI https://doi.org/10.1112/jlms/s2-6.3.437
- Allan J. Sieradski, A coloring test for asphericity, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 133, 97–106. MR 688427, DOI https://doi.org/10.1093/qmath/34.1.97
- John R. Stallings, Adian groups and pregroups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 321–342. MR 919831, DOI https://doi.org/10.1007/978-1-4613-9586-7_5
WiseNonPositiveCoherence02 Daniel T. Wise. Coherence, local indicability, and non-positive immersions. Preprint 2002.
WisePositiveRelatorCoherence02 Daniel T. Wise. Positive one-relator groups are coherent. Preprint 2002.
WiseSectional02 Daniel T. Wise. Sectional curvature and local quasiconvexity. Preprint 2002.
- Daniel T. Wise, Incoherent negatively curved groups, Proc. Amer. Math. Soc. 126 (1998), no. 4, 957–964. MR 1423338, DOI https://doi.org/10.1090/S0002-9939-98-04146-X
WiseNoCore Daniel T. Wise. A covering space with no compact core. Geom. Ded., 92(1):59–62, 2002.
BallmannBuyalo96 W. Ballmann and S. Buyalo. Nonpositively curved metrics on $2$-polyhedra. Math. Z., 222(1):97–134, 1996.
BaumslagProblems74 Gilbert Baumslag. Some problems on one-relator groups. In Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973), volume 372 of Lecture Notes in Math., pages 75–81. Springer, Berlin, 1974.
BridsonHaefliger Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature. Springer-Verlag, Berlin, 1999.
Brodskii80 S. D. Brodskiĭ. Equations over groups and groups with one defining relation. Uspekhi Mat. Nauk, 35(4(214)):183, 1980.
Brodskii84 S. D. Brodskiĭ. Equations over groups, and groups with one defining relation. Sibirsk. Mat. Zh., 25(2):84–103, 1984.
BurnsHale72 R. G. Burns and V. W. D. Hale. A note on group rings of certain torsion-free groups. Canad. Math. Bull., 15:441–445, 1972.
CorsonTrace2000 J. M. Corson and B. Trace. Diagrammatically reducible complexes and Haken manifolds. J. Austral. Math. Soc. Ser. A, 69(1):116–126, 2000.
Dicks94 Warren Dicks. Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. Invent. Math., 117(3):373–389, 1994.
EpsteinPenner88 D. B. A. Epstein and R. C. Penner. Euclidean decompositions of noncompact hyperbolic manifolds. J. Differential Geom., 27(1):67–80, 1988.
FeighnHandelCoherence Mark Feighn and Michael Handel. Mapping tori of free group automorphisms are coherent. Ann. of Math. (2), 149(3):1061–1077, 1999.
FischerKarrassSolitar72 J. Fischer, A. Karrass, and D. Solitar. On one-relator groups having elements of finite order. Proc. Amer. Math. Soc., 33:297–301, 1972.
GerstenReducible87 S. M. Gersten. Reducible diagrams and equations over groups. In Essays in group theory, pages 15–73. Springer, New York-Berlin, 1987.
Howie81 James Howie. On pairs of $2$-complexes and systems of equations over groups. J. Reine Angew. Math., 324:165–174, 1981.
Howie82 James Howie. On locally indicable groups. Math. Z., 180(4):445–461, 1982.
Howie87 James Howie. How to generalize one-relator group theory. In S. M. Gersten and John R. Stallings, editors, Combinatorial group theory and topology, pages 53–78, Princeton, N.J., 1987. Princeton Univ. Press.
Huebschmann81 Johannes Huebschmann. Aspherical $2$-complexes and an unsettled problem of J. H. C. Whitehead. Math. Ann., 258(1):17–37, 1981/82.
LS77 Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89.
Magnus30 Wilhelm Magnus. Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz). J. Reine Angew. Math., 163:141–165, 1930.
Magnus32 Wilhelm Magnus. Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann., 106:295–307, 1932.
McCammondWiseCoherence Jonathan P. McCammond and Daniel T. Wise. Coherence, local quasiconvexity and the perimeter of $2$-complexes. Preprint, 1999.
McCammondWiseFanLadder Jonathan P. McCammond and Daniel T. Wise. Fans and ladders in small cancellation theory. Proc. London Math. Soc. (3), 84(3):599–644, 2002.
WNeumann89 Walter D. Neumann. On intersections of finitely generated subgroups of free groups. In Groups—Canberra 1989, pages 161–170. Springer, Berlin, 1990.
Papakyriakopoulos57 C. D. Papakyriakopoulos. On Dehn’s lemma and the asphericity of knots. Ann. of Math. (2), 66:1–26, 1957.
Paulin91 Frédéric Paulin. Constructions of hyperbolic groups via hyperbolizations of polyhedra. In Group theory from a geometrical viewpoint (Trieste, 1990), pages 313–372. World Sci. Publishing, River Edge, NJ, 1991.
PrideHyperbolicComplexes88 Stephen J. Pride. Star-complexes, and the dependence problems for hyperbolic complexes. Glasgow Math. J., 30(2):155–170, 1988.
Rips82 E. Rips. Subgroups of small cancellation groups. Bull. London Math. Soc., 14(1):45–47, 1982.
Scott73 G. P. Scott. Finitely generated $3$-manifold groups are finitely presented. J. London Math. Soc. (2), 6:437–440, 1973.
Sieradski84 Allan J. Sieradski. A coloring test for asphericity. Quart. J. Math. Oxford Ser. (2), 34(133):97–106, 1983.
Stallings87 John R. Stallings. Adian groups and pregroups. In Essays in group theory, pages 321–342. Springer, New York, 1987.
WiseNonPositiveCoherence02 Daniel T. Wise. Coherence, local indicability, and non-positive immersions. Preprint 2002.
WisePositiveRelatorCoherence02 Daniel T. Wise. Positive one-relator groups are coherent. Preprint 2002.
WiseSectional02 Daniel T. Wise. Sectional curvature and local quasiconvexity. Preprint 2002.
Wise98 Daniel T. Wise. Incoherent negatively curved groups. Proc. Amer. Math. Soc., 126(4):957–964, 1998.
WiseNoCore Daniel T. Wise. A covering space with no compact core. Geom. Ded., 92(1):59–62, 2002.
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Additional Information
Daniel T. Wise
Affiliation:
Department of Mathematics, McGill University, Montreal, Quebec, CA H3A 2K6, Canada
MR Author ID:
604784
ORCID:
0000-0003-0128-1353
Email:
wise@math.mcgill.ca
Keywords:
Coherent groups,
nonpositive curvature,
one-relator groups
Received by editor(s):
October 21, 2002
Published electronically:
January 10, 2003
Additional Notes:
Research supported by grants from FCAR and NSERC
Communicated by:
Walter Neumann
Article copyright:
© Copyright 2003
American Mathematical Society
