Absolute profinite rigidity and hyperbolic geometry

M. R. Bridson; D. B. McReynolds; A. W. Reid; R. Spitler

doi:https://doi.org/10.4007/annals.2020.192.3.1

Abstract

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm {PSL}(2,\mathbb {Z}[\omega ])$ with $\omega ^2+\omega +1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in ${\rm {PSL}}(2,\mathbb {C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).

The Supplemental Magma code for this paper is available at the following location:
https://doi.org/10.4007/annals.2020.192.3.1.code

Show/hide bibliography for this article

[magma_calcs] $Go to document$ M. R. Bridson, D. B. McReynolds, A. W. Reid, and R. Spitler, Supplemental Magma code to accompany this paper, 2020.

Show bibtex

@misc{magma_calcs, author = {Bridson, Martin R. and McReynolds, David Ben and Reid, Alan W. and Spitler, Ryan}, title = {Supplemental {M}agma code to accompany this paper}, year = {2020}, note = {10 pp.}, doi = {10.4007/annals.2020.192.3.1.code}, url = {https://doi.org/10.4007/annals.2020.192.3.1.code}, }
[Ad] $Go to document$ C. C. Adams, "Noncompact hyperbolic $3$-orbifolds of small volume," in Topology ’90, de Gruyter, Berlin, 1992, vol. 1, pp. 1-15.

Show bibtex

@INCOLLECTION{Ad, author = {Adams, Colin C.}, title = {Noncompact hyperbolic {$3$}-orbifolds of small volume}, booktitle = {Topology '90}, venue = {{C}olumbus, {OH}, 1990}, series = {Ohio State Univ. Math. Res. Inst. Publ.}, volume = {1}, pages = {1--15}, publisher = {de Gruyter, Berlin}, year = {1992}, mrclass = {57M50}, mrnumber = {1184398}, mrreviewer = {Boris N. Apanasov}, doi = {10.1515/9783110857726}, url = {https://doi.org/10.1515/9783110857726}, zblnumber = {0801.57006}, }
[Alp] $Go to document$ R. C. Alperin, "Normal subgroups of ${ PSL}_2(\Bbb Z[\sqrt{-3}])$," Proc. Amer. Math. Soc., vol. 124, iss. 10, pp. 2935-2941, 1996.

Show bibtex

@ARTICLE{Alp, author = {Alperin, Roger C.}, title = {Normal subgroups of {${\rm PSL}_2(\bold Z[\sqrt{-3}])$}}, journal = {Proc. Amer. Math. Soc.}, fjournal = {Proceedings of the Amer. Math. Soc.}, volume = {124}, year = {1996}, number = {10}, pages = {2935--2941}, issn = {0002-9939}, mrclass = {20G30 (11F06 20H05)}, mrnumber = {1340373}, mrreviewer = {Alla S. Detinko}, doi = {10.1090/S0002-9939-96-03429-6}, url = {https://doi.org/10.1090/S0002-9939-96-03429-6}, zblnumber = {0858.20036}, }
[BaR] $Go to document$ M. D. Baker and A. W. Reid, "Arithmetic knots in closed 3-manifolds," J. Knot Theory Ramifications, vol. 11, iss. 6, pp. 903-920, 2002.

Show bibtex

@article{BaR, author = {Baker, M. D. and Reid, A. W.}, title = {Arithmetic knots in closed 3-manifolds}, note = {Special issue: Knots 2000 Korea, Vol. 3}, venue = {Yongpyong}, journal = {J. Knot Theory Ramifications}, fjournal = {Journal of Knot Theory and its Ramifications}, volume = {11}, year = {2002}, number = {6}, pages = {903--920}, issn = {0218-2165}, mrclass = {57M25 (57M50)}, mrnumber = {1936242}, mrreviewer = {Kerry N. Jones}, doi = {10.1142/S0218216502002049}, url = {https://doi.org/10.1142/S0218216502002049}, zblnumber = {1023.57008}, }
[Baum74] $Go to document$ G. Baumslag, "Residually finite groups with the same finite images," Compositio Math., vol. 29, iss. 3, pp. 249-252, 1974.

Show bibtex

@ARTICLE{Baum74, author = {Baumslag, Gilbert}, title = {Residually finite groups with the same finite images}, journal = {Compositio Math.}, fjournal = {Compositio Mathematica}, volume = {29}, year = {1974}, pages = {249--252}, issn = {0010-437X}, mrclass = {20E25}, mrnumber = {0357615}, mrreviewer = {A. H. Rhemtulla}, number = {3}, url = {http://www.numdam.org/item/CM_1974__29_3_249_0/}, zblnumber = {0309.20007}, }
[BF] $Go to document$ M. Boileau and S. Friedl, "The profinite completion of 3-manifold groups, fiberedness and the Thurston norm," in What’s Next?: The Mathematical Legacy of William P. Thurston, Princeton Univ. Press, Princeton, NJ, 2020, vol. 205, pp. 21-44.

Show bibtex

@incollection{BF, author = {Boileau, Michel and Friedl, Stefan}, title = {The profinite completion of 3-manifold groups, fiberedness and the {T}hurston norm}, booktitle={What's Next?: The Mathematical Legacy of William P. Thurston}, series = {Annals. Math. Stud.}, volume = {205}, pages = {21--44}, publisher = {Princeton Univ. Press, Princeton, NJ}, year = {2020}, doi = {10.1515/9780691185897-003}, url = {https://doi.org/10.1515/9780691185897-003}, zblnumber = {}, }
[Bos] $Go to document$ W. Bosma, J. Cannon, and C. Playoust, "The Magma algebra system. I. The user language," J. Symbolic Comput., vol. 24, iss. 3-4, pp. 235-265, 1997.

Show bibtex

@ARTICLE{Bos, author = {Bosma, Wieb and Cannon, John and Playoust, Catherine}, title = {The {M}agma algebra system. {I}. {T}he user language}, titlenote = {Computational Algebra and Number Theory (London, 1993)}, journal = {J. Symbolic Comput.}, fjournal = {Journal of Symbolic Computation}, volume = {24}, year = {1997}, number = {3-4}, pages = {235--265}, issn = {0747-7171}, mrclass = {68Q40}, mrnumber = {1484478}, doi = {10.1006/jsco.1996.0125}, url = {https://doi.org/10.1006/jsco.1996.0125}, zblnumber = {0898.68039}, }
[BZ] $Go to document$ S. Boyer and X. Zhang, "On Culler-Shalen seminorms and Dehn filling," Ann. of Math. (2), vol. 148, iss. 3, pp. 737-801, 1998.

Show bibtex

@ARTICLE{BZ, author = {Boyer, S. and Zhang, X.}, title = {On {C}uller-{S}halen seminorms and {D}ehn filling}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {148}, year = {1998}, number = {3}, pages = {737--801}, issn = {0003-486X}, mrclass = {57N10 (57M05 57M25 57M50)}, mrnumber = {1670053}, mrreviewer = {Kimihiko Motegi}, doi = {10.2307/121031}, url = {https://doi.org/10.2307/121031}, zblnumber = {1007.57016}, }
[BCR] $Go to document$ M. R. Bridson, M. D. E. Conder, and A. W. Reid, "Determining Fuchsian groups by their finite quotients," Israel J. Math., vol. 214, iss. 1, pp. 1-41, 2016.

Show bibtex

@ARTICLE{BCR, author = {Bridson, M. R. and Conder, M. D. E. and Reid, A. W.}, title = {Determining {F}uchsian groups by their finite quotients}, journal = {Israel J. Math.}, fjournal = {Israel Journal of Mathematics}, volume = {214}, year = {2016}, number = {1}, pages = {1--41}, issn = {0021-2172}, mrclass = {20E26 (20E18 20F36 20F67 20H10 20J06 22E40 57M07)}, mrnumber = {3540604}, mrreviewer = {Matthew Stover}, doi = {10.1007/s11856-016-1341-6}, url = {https://doi.org/10.1007/s11856-016-1341-6}, zblnumber = {1361.20037}, }
[BMRS2] M. R. Bridson, D. B. McReynolds, A. W. Reid, and R. Spitler, On the profinite rigidity of triangle groups, 2020.

Show bibtex

@MISC{BMRS2, author = {Bridson, M. R. and McReynolds, D. B. and Reid, A. W. and Spitler, R.}, title = {On the profinite rigidity of triangle groups}, year = {2020}, arxiv = {2004.07137}, zblnumber = {}, }
[BridR] $Go to document$ M. R. Bridson and A. W. Reid, "Profinite rigidity, fibering and the figure-eight knot," in What’s Next?: The Mathematical Legacy of William P. Thurston, Princeton Univ. Press, Princeton, NJ, 2020, vol. 205, pp. 45-64.

Show bibtex

@incollection{BridR, author = {Bridson, Martin R. and Reid, Alan W.}, title = {Profinite rigidity, fibering and the figure-eight knot}, booktitle = {What's Next?: The Mathematical Legacy of William P. Thurston}, series = {Annals. Math. Stud.}, volume = {205}, pages = {45--64}, publisher = {Princeton Univ. Press, Princeton, NJ}, year = {2020}, doi = {10.1515/9780691185897-004}, url = {https://10.1515/9780691185897-004}, zblnumber = {}, }
[BRW] $Go to document$ M. R. Bridson, A. W. Reid, and H. Wilton, "Profinite rigidity and surface bundles over the circle," Bull. Lond. Math. Soc., vol. 49, iss. 5, pp. 831-841, 2017.

Show bibtex

@ARTICLE{BRW, author = {Bridson, Martin R. and Reid, Alan W. and Wilton, Henry}, title = {Profinite rigidity and surface bundles over the circle}, journal = {Bull. Lond. Math. Soc.}, fjournal = {Bulletin of the London Mathematical Society}, volume = {49}, year = {2017}, number = {5}, pages = {831--841}, issn = {0024-6093}, mrclass = {57M27 (20E18 20E26)}, mrnumber = {3742450}, mrreviewer = {Steffen Kionke}, doi = {10.1112/blms.12076}, url = {https://doi.org/10.1112/blms.12076}, zblnumber = {06797282}, }
[BLW] $Go to document$ A. M. Brunner, Y. W. Lee, and N. J. Wielenberg, "Polyhedral groups and graph amalgamation products," Topology Appl., vol. 20, iss. 3, pp. 289-304, 1985.

Show bibtex

@ARTICLE{BLW, author = {Brunner, Andrew M. and Lee, Youn W. and Wielenberg, Norbert J.}, title = {Polyhedral groups and graph amalgamation products}, journal = {Topology Appl.}, fjournal = {Topology and its Applications}, volume = {20}, year = {1985}, number = {3}, pages = {289--304}, issn = {0166-8641}, mrclass = {57S30 (20E06 20F05 22E40)}, mrnumber = {0804041}, mrreviewer = {J. H. Rubinstein}, doi = {10.1016/0166-8641(85)90096-3}, url = {https://doi.org/10.1016/0166-8641(85)90096-3}, zblnumber = {0571.57032}, }
[Butt] $Go to document$ J. O. Button, "Fibred and virtually fibred hyperbolic 3-manifolds in the censuses," Experiment. Math., vol. 14, iss. 2, pp. 231-255, 2005.

Show bibtex

@ARTICLE{Butt, author = {Button, J. O.}, title = {Fibred and virtually fibred hyperbolic 3-manifolds in the censuses}, journal = {Experiment. Math.}, fjournal = {Experimental Mathematics}, volume = {14}, year = {2005}, number = {2}, pages = {231--255}, issn = {1058-6458}, mrclass = {57N10 (57M50)}, mrnumber = {2169525}, mrreviewer = {Colin C. Adams}, doi = {10.1080/10586458.2005.10128920}, url = {https://doi.org/10.1080/10586458.2005.10128920}, zblnumber = {1085.57012}, }
[CalDW] $Go to document$ P. J. Callahan, J. C. Dean, and J. R. Weeks, "The simplest hyperbolic knots," J. Knot Theory Ramifications, vol. 8, iss. 3, pp. 279-297, 1999.

Show bibtex

@ARTICLE{CalDW, author = {Callahan, Patrick J. and Dean, John C. and Weeks, Jeffrey R.}, title = {The simplest hyperbolic knots}, journal = {J. Knot Theory Ramifications}, fjournal = {Journal of Knot Theory and its Ramifications}, volume = {8}, year = {1999}, number = {3}, pages = {279--297}, issn = {0218-2165}, mrclass = {57M25 (57M50)}, mrnumber = {1691433}, mrreviewer = {Colin C. Adams}, doi = {10.1142/S0218216599000195}, url = {https://doi.org/10.1142/S0218216599000195}, zblnumber = {0933.57010}, }
[Knot] $Go to document$ J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, 2018.

Show bibtex

@MISC{Knot, author = {Cha, J. C. and Livingston, C.}, title = {Kno{t}Info: {T}able of Knot Invariants}, url = {http://www.indiana.edu/~knotinfo}, year = {2018}, zblnumber = {}, }
[CHLR] $Go to document$ T. Chinburg, E. Hamilton, D. D. Long, and A. W. Reid, "Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds," Duke Math. J., vol. 145, iss. 1, pp. 25-44, 2008.

Show bibtex

@ARTICLE{CHLR, author = {Chinburg, T. and Hamilton, E. and Long, D. D. and Reid, A. W.}, title = {Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds}, journal = {Duke Math. J.}, fjournal = {Duke Mathematical Journal}, volume = {145}, year = {2008}, number = {1}, pages = {25--44}, issn = {0012-7094}, mrclass = {58J53 (11R42 53C22 57M50)}, mrnumber = {2451288}, mrreviewer = {Colin C. Adams}, doi = {10.1215/00127094-2008-045}, url = {https://doi.org/10.1215/00127094-2008-045}, zblnumber = {1169.53030}, }
[snap] $Go to document$ D. Coulson, O. A. Goodman, C. D. Hodgson, and W. D. Neumann, "Computing arithmetic invariants of 3-manifolds," Experiment. Math., vol. 9, iss. 1, pp. 127-152, 2000.

Show bibtex

@ARTICLE{snap, author = {Coulson, David and Goodman, Oliver A. and Hodgson, Craig D. and Neumann, Walter D.}, title = {Computing arithmetic invariants of 3-manifolds}, journal = {Experiment. Math.}, fjournal = {Experimental Mathematics}, volume = {9}, year = {2000}, number = {1}, pages = {127--152}, issn = {1058-6458}, mrclass = {57M27 (11Y40 58J28)}, mrnumber = {1758805}, doi = {10.1080/10586458.2000.10504641}, url = {https://doi.org/10.1080/10586458.2000.10504641}, zblnumber = {1002.57044}, }
[CDW] $Go to document$ M. Culler, N. M. Dunfield, M. Goerner, and J. R. Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds.

Show bibtex

@MISC{CDW, author = {Culler, M. and Dunfield, N. M. and Goerner, M. and Weeks, J. R.}, title = {Snap{P}y, a computer program for studying the geometry and topology of 3-manifolds}, url = {http://snappy.computop.org}, zblnumber = {}, }
[DFPR] $Go to document$ J. D. Dixon, E. W. Formanek, J. C. Poland, and L. Ribes, "Profinite completions and isomorphic finite quotients," J. Pure Appl. Algebra, vol. 23, iss. 3, pp. 227-231, 1982.

Show bibtex

@ARTICLE{DFPR, author = {Dixon, John D. and Formanek, Edward W. and Poland, John C. and Ribes, Luis}, title = {Profinite completions and isomorphic finite quotients}, journal = {J. Pure Appl. Algebra}, fjournal = {Journal of Pure and Applied Algebra}, volume = {23}, year = {1982}, number = {3}, pages = {227--231}, issn = {0022-4049}, mrclass = {20E18}, mrnumber = {0644274}, mrreviewer = {S. P. Demushkin}, doi = {10.1016/0022-4049(82)90098-6}, url = {https://doi.org/10.1016/0022-4049(82)90098-6}, zblnumber = {0477.20014}, }
[DuMe] $Go to document$ W. D. Dunbar and R. G. Meyerhoff, "Volumes of hyperbolic $3$-orbifolds," Indiana Univ. Math. J., vol. 43, iss. 2, pp. 611-637, 1994.

Show bibtex

@ARTICLE{DuMe, author = {Dunbar, William D. and Meyerhoff, G. Robert}, title = {Volumes of hyperbolic {$3$}-orbifolds}, journal = {Indiana Univ. Math. J.}, fjournal = {Indiana University Mathematics Journal}, volume = {43}, year = {1994}, number = {2}, pages = {611--637}, issn = {0022-2518}, mrclass = {57M50 (30F40)}, mrnumber = {1291531}, mrreviewer = {Colin C. Adams}, doi = {10.1512/iumj.1994.43.43025}, url = {https://doi.org/10.1512/iumj.1994.43.43025}, zblnumber = {0899.57006}, }
[Fi] B. Fine, Algebraic theory of the Bianchi groups, Marcel Dekker, Inc., New York, 1989, vol. 129.

Show bibtex

@BOOK{Fi, author = {Fine, Benjamin}, title = {Algebraic theory of the {B}ianchi groups}, series = {Monographs and Textbooks in Pure and Applied Mathematics}, volume = {129}, publisher = {Marcel Dekker, Inc., New York}, year = {1989}, pages = {viii+249}, isbn = {0-8247-8192-9}, mrclass = {20-02 (11F06 20G30 22E40)}, mrnumber = {1010229}, mrreviewer = {C. Maclachlan}, zblnumber = {0760.20014}, }
[Fu] $Go to document$ L. Funar, "Torus bundles not distinguished by TQFT invariants," Geom. Topol., vol. 17, iss. 4, pp. 2289-2344, 2013.

Show bibtex

@ARTICLE{Fu, author = {Funar, Louis}, title = {Torus bundles not distinguished by {TQFT} invariants}, note = {with an appendix by Funar and Andrei Rapinchuk}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {17}, year = {2013}, number = {4}, pages = {2289--2344}, issn = {1465-3060}, mrclass = {57M27 (20H05 57R56)}, mrnumber = {3109869}, mrreviewer = {Matthew B. Day}, doi = {10.2140/gt.2013.17.2289}, url = {https://doi.org/10.2140/gt.2013.17.2289}, zblnumber = {1278.57020}, }
[GS] $Go to document$ F. J. Grunewald and R. Scharlau, "A note on finitely generated torsion-free nilpotent groups of class $2$," J. Algebra, vol. 58, iss. 1, pp. 162-175, 1979.

Show bibtex

@ARTICLE{GS, author = {Grunewald, Fritz J. and Scharlau, Rudolf}, title = {A note on finitely generated torsion-free nilpotent groups of class {$2$}}, journal = {J. Algebra}, fjournal = {Journal of Algebra}, volume = {58}, year = {1979}, number = {1}, pages = {162--175}, issn = {0021-8693}, mrclass = {20F18}, mrnumber = {0535851}, mrreviewer = {U. Stammbach}, doi = {10.1016/0021-8693(79)90197-2}, url = {https://doi.org/10.1016/0021-8693(79)90197-2}, zblnumber = {0406.20031}, }
[He] J. Hempel, Some 3-manifold groups with the same finite quotients, 2014.

Show bibtex

@MISC{He, author = {Hempel, J.}, title = {Some 3-manifold groups with the same finite quotients}, year = {2014}, arxiv = {1409.3509}, zblnumber = {}, }
[Iwasawa] $Go to document$ K. Iwasawa, "On the rings of valuation vectors," Ann. of Math. (2), vol. 57, pp. 331-356, 1953.

Show bibtex

@ARTICLE{Iwasawa, author = {Iwasawa, Kenkichi}, title = {On the rings of valuation vectors}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {57}, year = {1953}, pages = {331--356}, issn = {0003-486X}, mrclass = {10.0X}, mrnumber = {0053970}, mrreviewer = {J. T. Tate}, doi = {10.2307/1969863}, url = {https://doi.org/10.2307/1969863}, zblnumber = {0053.35603}, }
[Jan] G. J. Janusz, Algebraic Number Fields, Second ed., Amer. Math. Soc., Providence, RI, 1996, vol. 7.

Show bibtex

@BOOK{Jan, author = {Janusz, Gerald J.}, title = {Algebraic Number Fields}, series = {Grad. Stud. Math.}, volume = {7}, edition = {Second}, publisher = {Amer. Math. Soc., Providence, RI}, year = {1996}, pages = {x+276}, isbn = {0-8218-0429-4}, mrclass = {11Rxx (11-02 11R37)}, mrnumber = {1362545}, mrreviewer = {W. Narkiewicz}, zblnumber = {0854.11001}, }
[Kan] T. Kanenobu, "The augmentation subgroup of a pretzel link," Math. Sem. Notes Kobe Univ., vol. 7, iss. 2, pp. 363-384, 1979.

Show bibtex

@ARTICLE{Kan, author = {Kanenobu, Taizo}, title = {The augmentation subgroup of a pretzel link}, journal = {Math. Sem. Notes Kobe Univ.}, fjournal = {Kobe University. Mathematics Seminar Notes}, volume = {7}, year = {1979}, number = {2}, pages = {363--384}, issn = {0385-633x}, mrclass = {57M25}, mrnumber = {0557309}, mrreviewer = {Kenneth A. Perko, Jr.}, zblnumber = {0456.57002}, }
[MR] $Go to document$ C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, Springer-Verlag, New York, 2003, vol. 219.

Show bibtex

@BOOK{MR, author = {Maclachlan, Colin and Reid, Alan W.}, title = {The Arithmetic of Hyperbolic 3-Manifolds}, series = {Grad. Texts Math.}, volume = {219}, publisher = {Springer-Verlag, New York}, year = {2003}, pages = {xiv+463}, isbn = {0-387-98386-4}, mrclass = {57M50 (11R52)}, mrnumber = {1937957}, mrreviewer = {Kerry N. Jones}, doi = {10.1007/978-1-4757-6720-9}, url = {https://doi.org/10.1007/978-1-4757-6720-9}, zblnumber = {1025.57001}, }
[Mar] $Go to document$ D. A. Marcus, Number Fields, Springer, Cham, 2018.

Show bibtex

@BOOK{Mar, author = {Marcus, Daniel A.}, title = {Number Fields}, series = {Universitext}, note = {Second edition of \mr{0457396}, with a foreword by Barry Mazur}, publisher = {Springer, Cham}, year = {2018}, pages = {xviii+203}, isbn = {978-3-319-90232-6; 978-3-319-90233-3}, mrclass = {11-01 (11Rxx 11Txx 12-01)}, mrnumber = {3822326}, doi = {10.1007/978-3-319-90233-3}, url = {https://doi.org/10.1007/978-3-319-90233-3}, zblnumber = {1411.11003}, }
[Rem] The Kourovka Notebook. Unsolved Problems in Group Theory, Including Archive of Solved Problems, Seventeenth ed., Mazurov, V. D. and Khukhro, E. I., Eds., Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 2010.

Show bibtex

@BOOK{Rem, title = {The {K}ourovka Notebook. Unsolved Problems in Group Theory, Including Archive of Solved Problems}, edition = {Seventeenth}, editor = {Mazurov, V. D. and Khukhro, E. I.}, publisher = {Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk}, year = {2010}, pages = {139}, mrclass = {20-02 (00A07)}, mrnumber = {3235009}, zblnumber = {1211.20001}, }
[McR] $Go to document$ D. B. McReynolds, "Peripheral separability and cusps of arithmetic hyperbolic orbifolds," Algebr. Geom. Topol., vol. 4, pp. 721-755, 2004.

Show bibtex

@ARTICLE{McR, author = {McReynolds, D. B.}, title = {Peripheral separability and cusps of arithmetic hyperbolic orbifolds}, journal = {Algebr. Geom. Topol.}, fjournal = {Algebraic \& Geometric Topology}, volume = {4}, year = {2004}, pages = {721--755}, issn = {1472-2747}, mrclass = {57M50 (11F06 20E26 30F40 57N16 57R90)}, mrnumber = {2100678}, mrreviewer = {Kerry N. Jones}, doi = {10.2140/agt.2004.4.721}, url = {https://doi.org/10.2140/agt.2004.4.721}, zblnumber = {1058.57012}, }
[McSpit] D. B. McReynolds and R. Spitler, Profinite completions of linear groups and rigid representation.

Show bibtex

@MISC{McSpit, author = {McReynolds, D. B. and Spitler, R.}, title = {Profinite completions of linear groups and rigid representation}, note = {in preparation}, zblnumber = {}, }
[MedV] $Go to document$ A. Mednykh and A. Vesnin, "Visualization of the isometry group action on the Fomenko-Matveev-Weeks manifold," J. Lie Theory, vol. 8, iss. 1, pp. 51-66, 1998.

Show bibtex

@ARTICLE{MedV, author = {Mednykh, Alexander and Vesnin, Andrei}, title = {Visualization of the isometry group action on the {F}omenko-{M}atveev-{W}eeks manifold}, journal = {J. Lie Theory}, fjournal = {Journal of Lie Theory}, volume = {8}, year = {1998}, number = {1}, pages = {51--66}, issn = {0949-5932}, mrclass = {57M60 (57M12 57N10)}, mrnumber = {1616790}, url = {https://www.emis.de/journals/JLT/vol.8_no.1/3.html}, zblnumber = {0891.57014}, }
[Mey] $Go to document$ R. Meyerhoff, "The cusped hyperbolic $3$-orbifold of minimum volume," Bull. Amer. Math. Soc. (N.S.), vol. 13, iss. 2, pp. 154-156, 1985.

Show bibtex

@ARTICLE{Mey, author = {Meyerhoff, Robert}, title = {The cusped hyperbolic {$3$}-orbifold of minimum volume}, journal = {Bull. Amer. Math. Soc. (N.S.)}, fjournal = {Amer. Math. Soc.. Bulletin. New Series}, volume = {13}, year = {1985}, number = {2}, pages = {154--156}, issn = {0273-0979}, mrclass = {22E40 (51M25 57M25 57S30)}, mrnumber = {0799800}, mrreviewer = {William Harvey}, doi = {10.1090/S0273-0979-1985-15401-1}, url = {https://doi.org/10.1090/S0273-0979-1985-15401-1}, zblnumber = {0602.57009}, }
[NS] $Go to document$ N. Nikolov and D. Segal, "On finitely generated profinite groups. I. Strong completeness and uniform bounds," Ann. of Math. (2), vol. 165, iss. 1, pp. 171-238, 2007.

Show bibtex

@ARTICLE{NS, author = {Nikolov, Nikolay and Segal, Dan}, title = {On finitely generated profinite groups. {I}. {S}trong completeness and uniform bounds}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {165}, year = {2007}, number = {1}, pages = {171--238}, issn = {0003-486X}, mrclass = {20E18 (20E32 20F12)}, mrnumber = {2276769}, mrreviewer = {Benjamin Klopsch}, doi = {10.4007/annals.2007.165.171}, url = {https://doi.org/10.4007/annals.2007.165.171}, zblnumber = {1126.20018}, }
[PZ] $Go to document$ L. Paoluzzi and B. Zimmermann, "Finite quotients of the Picard group and related hyperbolic tetrahedral and Bianchi groups," Rend. Istit. Mat. Univ. Trieste, vol. 32, iss. suppl. 1, pp. 257-288, 2001.

Show bibtex

@ARTICLE{PZ, author = {Paoluzzi, Luisa and Zimmermann, Bruno}, title = {Finite quotients of the {P}icard group and related hyperbolic tetrahedral and {B}ianchi groups}, note = {dedicated to the memory of Marco Reni}, journal = {Rend. Istit. Mat. Univ. Trieste}, fjournal = {Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics}, volume = {32}, year = {2001}, number = {suppl. 1}, pages = {257--288}, issn = {0049-4704}, mrclass = {20H10 (11F06 57M50)}, mrnumber = {1893401}, mrreviewer = {Leonid Vulakh}, url = {https://rendiconti.dmi.units.it/volumi/32s1/14.pdf}, zblnumber = {1006.11018}, }
[Per] $Go to document$ R. Perlis, "On the equation $\zeta_{K}(s)=\zeta_{K’}(s)$," J. Number Theory, vol. 9, iss. 3, pp. 342-360, 1977.

Show bibtex

@ARTICLE{Per, author = {Perlis, Robert}, title = {On the equation {$\zeta_{K}(s)=\zeta_{K'}(s)$}}, journal = {J. Number Theory}, fjournal = {Journal of Number Theory}, volume = {9}, year = {1977}, number = {3}, pages = {342--360}, issn = {0022-314X}, mrclass = {12A70 (20D10)}, mrnumber = {0447188}, mrreviewer = {B. Z. Moroz}, doi = {10.1016/0022-314X(77)90070-1}, url = {https://doi.org/10.1016/0022-314X(77)90070-1}, zblnumber = {0389.12006}, }
[Rag] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, Berlin, 1972, vol. 68.

Show bibtex

@BOOK{Rag, author = {Raghunathan, M. S.}, title = {Discrete Subgroups of {L}ie Groups}, note = {Ergeb. Math. Grenzgebi.}, volume = {68}, publisher = {Springer-Verlag, Berlin}, year = {1972}, pages = {ix+227}, mrclass = {22E40}, mrnumber = {0507234}, mrreviewer = {J. S. Joel}, zblnumber = {0254.22005}, }
[ReWan] $Go to document$ A. W. Reid and S. Wang, "Non-Haken $3$-manifolds are not large with respect to mappings of non-zero degree," Comm. Anal. Geom., vol. 7, iss. 1, pp. 105-132, 1999.

Show bibtex

@ARTICLE{ReWan, author = {Reid, Alan W. and Wang, Shicheng}, title = {Non-{H}aken {$3$}-manifolds are not large with respect to mappings of non-zero degree}, journal = {Comm. Anal. Geom.}, fjournal = {Communications in Analysis and Geometry}, volume = {7}, year = {1999}, number = {1}, pages = {105--132}, issn = {1019-8385}, mrclass = {57N10 (57M05 57M07 57M50)}, mrnumber = {1674109}, mrreviewer = {Tsuyoshi Kobayashi}, doi = {10.4310/CAG.1999.v7.n1.a4}, url = {https://doi.org/10.4310/CAG.1999.v7.n1.a4}, zblnumber = {0930.57012}, }
[RZ] $Go to document$ L. Ribes and P. Zalesskii, Profinite Groups, Springer-Verlag, Berlin, 2000, vol. 40.

Show bibtex

@BOOK{RZ, author = {Ribes, Luis and Zalesskii, Pavel}, title = {Profinite Groups}, series = {Ergeb. Math. Grenzgeb.}, volume = {40}, publisher = {Springer-Verlag, Berlin}, year = {2000}, pages = {xiv+435}, isbn = {3-540-66986-8}, mrclass = {20E18}, mrnumber = {1775104}, mrreviewer = {Marcus du Sautoy}, doi = {10.1007/978-3-662-04097-3}, url = {https://doi.org/10.1007/978-3-662-04097-3}, zblnumber = {0949.20017}, }
[Scott] $Go to document$ G. P. Scott, "Compact submanifolds of $3$-manifolds," J. London Math. Soc. (2), vol. 7, pp. 246-250, 1973.

Show bibtex

@ARTICLE{Scott, author = {Scott, G. P.}, title = {Compact submanifolds of {$3$}-manifolds}, journal = {J. London Math. Soc. (2)}, fjournal = {Journal of the London Mathematical Society. Second Series}, volume = {7}, year = {1973}, pages = {246--250}, issn = {0024-6107}, mrclass = {57A10}, mrnumber = {0326737}, mrreviewer = {C. D. Feustel}, doi = {10.1112/jlms/s2-7.2.246}, url = {https://doi.org/10.1112/jlms/s2-7.2.246}, zblnumber = {0266.57001}, }
[segal] $Go to document$ D. Segal, Polycyclic Groups, Cambridge Univ. Press, Cambridge, 1983, vol. 82.

Show bibtex

@BOOK{segal, author = {Segal, Daniel}, title = {Polycyclic Groups}, series = {Cambridge Tracts in Math.}, volume = {82}, publisher = {Cambridge Univ. Press, Cambridge}, year = {1983}, pages = {xiv+289}, isbn = {0-521-24146-4}, mrclass = {20-02 (20F16)}, mrnumber = {0713786}, mrreviewer = {John F. Bowers}, doi = {10.1017/CBO9780511565953}, url = {https://doi.org/10.1017/CBO9780511565953}, zblnumber = {0516.20001}, }
[Serre] $Go to document$ . J-P. Serre, Trees, Springer-Verlag, New York, 1980.

Show bibtex

@BOOK{Serre, author = {Serre, {\relax J-P}}, title = {Trees}, note = {translated from the French by John Stillwell}, publisher = {Springer-Verlag, New York}, year = {1980}, pages = {ix+142}, isbn = {3-540-10103-9}, mrclass = {20H10 (05C05 22E50)}, mrnumber = {0607504}, zblnumber = {0548.20018}, doi = {10.1007/978-3-642-61856-7}, url = {https://www.doi.org/10.1007/978-3-642-61856-7}, }
[Spitler] $Go to document$ R. F. Spitler, Profinite Completions and Representations of Finitely Generated Groups, 2019.

Show bibtex

@MISC{Spitler, author = {Spitler, Ryan F.}, title = {Profinite Completions and Representations of Finitely Generated Groups}, note = {Ph.D. thesis, Purdue University}, year = {2019}, zblnumber = {}, doi = {10.25394/pgs.9117068.v1}, url = {https://www.doi.org/10.25394/pgs.9117068.v1}, }
[Vin] $Go to document$ È. B. Vinberg, "The smallest field of definition of a subgroup of the group ${ PSL}_2$," Mat. Sb., vol. 184, iss. 10, pp. 53-66, 1993.

Show bibtex

@ARTICLE{Vin, author = {Vinberg, {È}. B.}, title = {The smallest field of definition of a subgroup of the group {${\rm PSL}_2$}}, journal = {Mat. Sb.}, fjournal = {Matematicheskiĭ Sbornik}, volume = {184}, year = {1993}, number = {10}, pages = {53--66}, issn = {0368-8666}, mrclass = {20G20}, mrnumber = {1249413}, mrreviewer = {Ja. N. Nuzhin}, doi = {10.1070/SM1995v080n01ABEH003519}, url = {https://doi.org/10.1070/SM1995v080n01ABEH003519}, zblnumber = {0831.20061}, }
[Wei] $Go to document$ B. Weisfeiler, "Strong approximation for Zariski-dense subgroups of semisimple algebraic groups," Ann. of Math. (2), vol. 120, iss. 2, pp. 271-315, 1984.

Show bibtex

@ARTICLE{Wei, author = {Weisfeiler, Boris}, title = {Strong approximation for {Z}ariski-dense subgroups of semisimple algebraic groups}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {120}, year = {1984}, number = {2}, pages = {271--315}, issn = {0003-486X}, mrclass = {20G15 (11R32 11S75 20G30)}, mrnumber = {0763908}, mrreviewer = {James E. Humphreys}, doi = {10.2307/2006943}, url = {https://doi.org/10.2307/2006943}, zblnumber = {0568.14025}, }
[Wilk] $Go to document$ G. Wilkes, "Profinite rigidity for Seifert fibre spaces," Geom. Dedicata, vol. 188, pp. 141-163, 2017.

Show bibtex

@ARTICLE{Wilk, author = {Wilkes, Gareth}, title = {Profinite rigidity for {S}eifert fibre spaces}, journal = {Geom. Dedicata}, fjournal = {Geometriae Dedicata}, volume = {188}, year = {2017}, pages = {141--163}, issn = {0046-5755}, mrclass = {57M27 (57M05)}, mrnumber = {3639628}, mrreviewer = {Mark F. Hagen}, doi = {10.1007/s10711-016-0209-6}, url = {https://doi.org/10.1007/s10711-016-0209-6}, zblnumber = {1369.57003}, }
[WZ] $Go to document$ H. Wilton and P. Zalesskii, "Distinguishing geometries using finite quotients," Geom. Topol., vol. 21, iss. 1, pp. 345-384, 2017.

Show bibtex

@ARTICLE{WZ, author = {Wilton, Henry and Zalesskii, Pavel}, title = {Distinguishing geometries using finite quotients}, journal = {Geom. Topol.}, fjournal = {Geometry \& Topology}, volume = {21}, year = {2017}, number = {1}, pages = {345--384}, issn = {1465-3060}, mrclass = {20F65 (20E26 20F67 57M05 57N10)}, mrnumber = {3608716}, mrreviewer = {Thomas Koberda}, doi = {10.2140/gt.2017.21.345}, url = {https://doi.org/10.2140/gt.2017.21.345}, zblnumber = {1361.57023}, }

Absolute profinite rigidity and hyperbolic geometry

Abstract

Authors