Topological symmetry groups of graphs in $3$-manifolds
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- by Erica Flapan and Harry Tamvakis PDF
- Proc. Amer. Math. Soc. 141 (2013), 1423-1436 Request permission
Abstract:
We prove that for every closed, connected, orientable, irreducible 3-manifold there exists an alternating group $A_n$ which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group $G$ there is an embedding $\Gamma$ of some graph in a hyperbolic rational homology 3-sphere such that the topological symmetry group of $\Gamma$ is isomorphic to $G$.References
- Francis Bonahon, Geometric structures on 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 93–164. MR 1886669
- D. Cooper and D. D. Long, Free actions of finite groups on rational homology $3$-spheres, Topology Appl. 101 (2000), no. 2, 143–148. MR 1732066, DOI 10.1016/S0166-8641(98)00116-3
- Jonathan Dinkelbach and Bernhard Leeb, Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds, Geom. Topol. 13 (2009), no. 2, 1129–1173. MR 2491658, DOI 10.2140/gt.2009.13.1129
- Erica Flapan, Rigidity of graph symmetries in the $3$-sphere, J. Knot Theory Ramifications 4 (1995), no. 3, 373–388. MR 1347360, DOI 10.1142/S0218216595000181
- Erica Flapan, Ramin Naimi, James Pommersheim, and Harry Tamvakis, Topological symmetry groups of graphs embedded in the 3-sphere, Comment. Math. Helv. 80 (2005), no. 2, 317–354. MR 2142245, DOI 10.4171/CMH/16
- R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6 (1939), 239–250 (German). MR 1557026
- A. Hurwitz, Über algebraische Gebilde mit Eindeutigen Transformationen in sich, Math. Annalen 41 (1893), 403–442.
- William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
- Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
- John Kalliongis and Andy Miller, Geometric group actions on lens spaces, Kyungpook Math. J. 42 (2002), no. 2, 313–344. MR 1942194
- Sadayoshi Kojima, Bounding finite groups acting on $3$-manifolds, Math. Proc. Cambridge Philos. Soc. 96 (1984), no. 2, 269–281. MR 757660, DOI 10.1017/S0305004100062162
- Darryl McCullough, Isometries of elliptic 3-manifolds, J. London Math. Soc. (2) 65 (2002), no. 1, 167–182. MR 1875143, DOI 10.1112/S0024610701002782
- William H. Meeks III and Peter Scott, Finite group actions on $3$-manifolds, Invent. Math. 86 (1986), no. 2, 287–346. MR 856847, DOI 10.1007/BF01389073
- Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059, DOI 10.1007/978-1-4612-9906-6
- John W. Morgan and Frederick Tsz-Ho Fong, Ricci flow and geometrization of 3-manifolds, University Lecture Series, vol. 53, American Mathematical Society, Providence, RI, 2010. MR 2597148, DOI 10.1090/ulect/053
- John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563, DOI 10.1305/ndjfl/1193667709
- J. Morgan and G. Tian, Completion of the proof of the Geometrization Conjecture, arXiv:0809.4040
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
- Ken’ichi Ohshika, Teichmüller spaces of Seifert fibered manifolds with infinite $\pi _1$, Topology Appl. 27 (1987), no. 1, 75–93. MR 910495, DOI 10.1016/0166-8641(87)90058-7
- Jonathan Simon, Topological chirality of certain molecules, Topology 25 (1986), no. 2, 229–235. MR 837623, DOI 10.1016/0040-9383(86)90041-8
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
- Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
- Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594
- Bruno Zimmermann, On finite simple groups acting on homology 3-spheres, Topology Appl. 125 (2002), no. 2, 199–202. MR 1933571, DOI 10.1016/S0166-8641(01)00271-1
Additional Information
- Erica Flapan
- Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711
- Email: eflapan@pomona.edu
- Harry Tamvakis
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Email: harryt@math.umd.edu
- Received by editor(s): June 2, 2011
- Received by editor(s) in revised form: August 3, 2011
- Published electronically: August 3, 2012
- Additional Notes: The first author was supported in part by NSF Grant DMS-0905087.
The second author was supported in part by NSF Grant DMS-0901341. - Communicated by: Daniel Ruberman
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1423-1436
- MSC (2010): Primary 57M15, 57M60; Secondary 05C10, 05C25
- DOI: https://doi.org/10.1090/S0002-9939-2012-11405-4
- MathSciNet review: 3008889