Abstract
The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group, then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
-
[ArzhantsevaSteenbock14] G. Arzhantseva and M. Steenbock, Rips construction without unique product, 2014.
@MISC{ArzhantsevaSteenbock14,
author = {Arzhantseva, G. and Steenbock, M.},
title = {Rips construction without unique product},
year = {2014},
arxiv = {1407.2441},
zblnumber = {},
} -
[BartelsLueck12]
A. Bartels and W. Lück, "The Borel conjecture for hyperbolic and ${ CAT}(0)$-groups," Ann. of Math. (2), vol. 175, iss. 2, pp. 631-689, 2012.@ARTICLE{BartelsLueck12,
author = {Bartels, Arthur and Lück, Wolfgang},
title = {The {B}orel conjecture for hyperbolic and {${\rm CAT}(0)$}-groups},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {175},
year = {2012},
number = {2},
pages = {631--689},
issn = {0003-486X},
mrclass = {57M07 (20F67)},
mrnumber = {2993750},
mrreviewer = {Paul D. Mitchener},
doi = {10.4007/annals.2012.175.2.5},
url = {https://doi.org/10.4007/annals.2012.175.2.5},
zblnumber = {1256.57021},
} -
[BartelsLueckReich08] A. Bartels, W. Lück, and H. Reich, "On the Farrell–Jones conjecture and its applications," J. Topol., vol. 1, iss. 1, pp. 57-86, 2008.
@ARTICLE{BartelsLueckReich08,
author = {Bartels, Arthur and Lück, Wolfgang and Reich, Holger},
title = {On the {F}arrell--{J}ones conjecture and its applications},
journal = {J. Topol.},
fjournal = {Journal of Topology},
volume = {1},
year = {2008},
number = {1},
pages = {57--86},
issn = {1753-8416},
mrclass = {19D55 (19A31 19B28)},
mrnumber = {2365652},
mrreviewer = {A. A. Ranicki},
doi = {10.1112/jtopol/jtm008},
url = {https://doi.org/10.1112/jtopol/jtm008},
zblnumber = {1141.19002},
} -
[Bowditch00] B. H. Bowditch, "A variation on the unique product property," J. London Math. Soc. (2), vol. 62, iss. 3, pp. 813-826, 2000.
@ARTICLE{Bowditch00,
author = {Bowditch, B. H.},
title = {A variation on the unique product property},
journal = {J. London Math. Soc. (2)},
fjournal = {Journal of the London Mathematical Society. Second Series},
volume = {62},
year = {2000},
number = {3},
pages = {813--826},
issn = {0024-6107},
mrclass = {20F65},
mrnumber = {1794287},
mrreviewer = {Thomas Delzant},
doi = {10.1112/S0024610700001307},
url = {https://doi.org/10.1112/S0024610700001307},
zblnumber = {1033.20040},
} -
[Carter14] W. Carter, "New examples of torsion-free non-unique product groups," J. Group Theory, vol. 17, iss. 3, pp. 445-464, 2014.
@ARTICLE{Carter14,
author = {Carter, William},
title = {New examples of torsion-free non-unique product groups},
journal = {J. Group Theory},
fjournal = {Journal of Group Theory},
volume = {17},
year = {2014},
number = {3},
pages = {445--464},
issn = {1433-5883},
mrclass = {20F99},
mrnumber = {3200369},
mrreviewer = {E. Formanek},
doi = {10.1515/jgt-2013-0051},
url = {https://doi.org/10.1515/jgt-2013-0051},
zblnumber = {1300.20041},
} -
[Cliff80] G. H. Cliff, "Zero divisors and idempotents in group rings," Canadian J. Math., vol. 32, iss. 3, pp. 596-602, 1980.
@ARTICLE{Cliff80,
author = {Cliff, Gerald H.},
title = {Zero divisors and idempotents in group rings},
journal = {Canadian J. Math.},
fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
volume = {32},
year = {1980},
number = {3},
pages = {596--602},
issn = {0008-414X},
mrclass = {16A26 (20C07)},
mrnumber = {0586978},
mrreviewer = {E. Formanek},
doi = {10.4153/CJM-1980-046-3},
url = {https://doi.org/10.4153/CJM-1980-046-3},
zblnumber = {0439.16011},
} -
[CravenPappas13] D. A. Craven and P. Pappas, "On the unit conjecture for supersoluble group algebras," J. Algebra, vol. 394, pp. 310-356, 2013.
@ARTICLE{CravenPappas13,
author = {Craven, David A. and Pappas, Peter},
title = {On the unit conjecture for supersoluble group algebras},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {394},
year = {2013},
pages = {310--356},
issn = {0021-8693},
mrclass = {16S34 (16U60)},
mrnumber = {3092724},
mrreviewer = {Zhengxing Li},
doi = {10.1016/j.jalgebra.2013.07.014},
url = {https://doi.org/10.1016/j.jalgebra.2013.07.014},
zblnumber = {1339.16037},
} -
[DykemaHeisterJuschenko15] K. Dykema, T. Heister, and K. Juschenko, "Finitely presented groups related to Kaplansky’s direct finiteness conjecture," Exp. Math., vol. 24, iss. 3, pp. 326-338, 2015.
@ARTICLE{DykemaHeisterJuschenko15,
author = {Dykema, Ken and Heister, Timo and Juschenko, Kate},
title = {Finitely presented groups related to {K}aplansky's direct finiteness conjecture},
journal = {Exp. Math.},
fjournal = {Experimental Mathematics},
volume = {24},
year = {2015},
number = {3},
pages = {326--338},
issn = {1058-6458},
mrclass = {16S34 (20C07)},
mrnumber = {3359220},
mrreviewer = {Donald S. Passman},
doi = {10.1080/10586458.2014.993051},
url = {https://doi.org/10.1080/10586458.2014.993051},
zblnumber = {1403.20004},
} -
[FarrellJones93] F. T. Farrell and L. E. Jones, "Isomorphism conjectures in algebraic $K$-theory," J. Amer. Math. Soc., vol. 6, iss. 2, pp. 249-297, 1993.
@ARTICLE{FarrellJones93,
author = {Farrell, F. T. and Jones, L. E.},
title = {Isomorphism conjectures in algebraic {$K$}-theory},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {6},
year = {1993},
number = {2},
pages = {249--297},
issn = {0894-0347},
mrclass = {57N37 (19D55)},
mrnumber = {1179537},
mrreviewer = {Masayuki Yamasaki},
doi = {10.2307/2152801},
url = {https://doi.org/10.2307/2152801},
zblnumber = {0798.57018},
} -
[Gardam21+] G. Gardam, Solving semidecidable problems in group theory, 2021.
@MISC{Gardam21+,
author = {Gardam, G.},
title = {Solving semidecidable problems in group theory},
year = {2021},
note = {in preparation},
zblnumber = {},
} -
[GruberMartinSteenbock15] D. Gruber, A. Martin, and M. Steenbock, "Finite index subgroups without unique product in graphical small cancellation groups," Bull. Lond. Math. Soc., vol. 47, iss. 4, pp. 631-638, 2015.
@ARTICLE{GruberMartinSteenbock15,
author = {Gruber, D. and Martin, A. and Steenbock, M.},
title = {Finite index subgroups without unique product in graphical small cancellation groups},
journal = {Bull. Lond. Math. Soc.},
fjournal = {Bulletin of the London Mathematical Society},
volume = {47},
year = {2015},
number = {4},
pages = {631--638},
issn = {0024-6093},
mrclass = {20F06 (20F67)},
mrnumber = {3375930},
mrreviewer = {Rémi Bernard Coulon},
doi = {10.1112/blms/bdv040},
url = {https://doi.org/10.1112/blms/bdv040},
zblnumber = {1337.20034},
} -
[Higman40b] G. Higman, Units in group rings, 1940.
@MISC{Higman40b,
author = {Higman, Graham},
title = {Units in group rings},
note = {D.Phil. thesis, University of Oxford},
year = {1940},
zblnumber = {},
} -
[Higman40a] G. Higman, "The units of group-rings," Proc. London Math. Soc. (2), vol. 46, pp. 231-248, 1940.
@ARTICLE{Higman40a,
author = {Higman, Graham},
title = {The units of group-rings},
journal = {Proc. London Math. Soc. (2)},
fjournal = {Proceedings of the London Mathematical Society. Second Series},
volume = {46},
year = {1940},
pages = {231--248},
issn = {0024-6115},
mrclass = {20.0X},
mrnumber = {0002137},
mrreviewer = {M. Hall},
doi = {10.1112/plms/s2-46.1.231},
url = {https://doi.org/10.1112/plms/s2-46.1.231},
zblnumber = {0025.24302},
} -
[HigsonKasparov01] N. Higson and G. Kasparov, "$E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space," Invent. Math., vol. 144, iss. 1, pp. 23-74, 2001.
@ARTICLE{HigsonKasparov01,
author = {Higson, Nigel and Kasparov, Gennadi},
title = {{$E$}-theory and {$KK$}-theory for groups which act properly and isometrically on {H}ilbert space},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {144},
year = {2001},
number = {1},
pages = {23--74},
issn = {0020-9910},
mrclass = {19K35 (19L47 46L80)},
mrnumber = {1821144},
mrreviewer = {Emmanuel C. Germain},
doi = {10.1007/s002220000118},
url = {https://doi.org/10.1007/s002220000118},
zblnumber = {0988.19003},
} -
[KammeyerLueckRueping16] H. Kammeyer, W. Lück, and H. Rüping, "The Farrell-Jones conjecture for arbitrary lattices in virtually connected Lie groups," Geom. Topol., vol. 20, iss. 3, pp. 1275-1287, 2016.
@ARTICLE{KammeyerLueckRueping16,
author = {Kammeyer, Holger and Lück, Wolfgang and Rüping, Henrik},
title = {The {F}arrell-{J}ones conjecture for arbitrary lattices in virtually connected {L}ie groups},
journal = {Geom. Topol.},
fjournal = {Geometry \& Topology},
volume = {20},
year = {2016},
number = {3},
pages = {1275--1287},
issn = {1465-3060},
mrclass = {19G24 (18F25 19A31 19B28 22E40)},
mrnumber = {3523058},
mrreviewer = {Jonathan M. Rosenberg},
doi = {10.2140/gt.2016.20.1275},
url = {https://doi.org/10.2140/gt.2016.20.1275},
zblnumber = {1346.18019},
} -
[Kaplansky57] I. Kaplansky, "Problems in the theory of rings," in Report of a Conference on Linear Algebras, National Academy of Sciences-National Research Council, Washington, Publ. 502, 1957, p. v.
@INCOLLECTION{Kaplansky57,
author = {Kaplansky, Irving},
title = {Problems in the theory of rings},
booktitle = {{R}eport of a Conference on Linear Algebras},
year = {1957},
pages = {v+60},
publisher = {National Academy of Sciences-National Research Council, Washington, Publ. 502},
mrclass = {16.00},
mrnumber = {0096696},
mrreviewer = {I. N. Herstein},
zblnumber = {0095.25602},
} -
[Kaplansky70] I. Kaplansky, "“Problems in the theory of rings” revisited," Amer. Math. Monthly, vol. 77, pp. 445-454, 1970.
@ARTICLE{Kaplansky70,
author = {Kaplansky, Irving},
title = {``{P}roblems in the theory of rings'' revisited},
journal = {Amer. Math. Monthly},
fjournal = {American Mathematical Monthly},
volume = {77},
year = {1970},
pages = {445--454},
issn = {0002-9890},
mrclass = {16.00},
mrnumber = {0258865},
doi = {10.2307/2317376},
url = {https://doi.org/10.2307/2317376},
zblnumber = {0208.29701},
} -
[Kourovka18] The Kourovka Notebook, Khukhro, E. I. and Mazurov, V. D., Eds., Sobolev Institute of Mathematics. Russian Academy of Sciences. Siberian Branch, Novosibirsk, 2018.
@BOOK{Kourovka18, title = {The {K}ourovka Notebook},
editor = {Khukhro, E. I. and Mazurov, V. D.},
note = {Unsolved problems in group theory, Nineteenth edition [\mr{0204500}], March 2019 update},
publisher = {Sobolev Institute of Mathematics. Russian Academy of Sciences. Siberian Branch, Novosibirsk},
year = {2018},
pages = {248},
mrclass = {20-02 (00A07)},
mrnumber = {3981599},
zblnumber = {},
} -
[KionkeRaimbault16] S. Kionke and J. Raimbault, "On geometric aspects of diffuse groups," Doc. Math., vol. 21, pp. 873-915, 2016.
@ARTICLE{KionkeRaimbault16,
author = {Kionke, Steffen and Raimbault, Jean},
title = {On geometric aspects of diffuse groups},
note = {with an appendix by Nathan Dunfield},
journal = {Doc. Math.},
fjournal = {Documenta Mathematica},
volume = {21},
year = {2016},
pages = {873--915},
issn = {1431-0635},
mrclass = {20F65 (20-04 20H15 22E40 57M07)},
mrnumber = {3548136},
mrreviewer = {Nansen Petrosyan},
zblnumber = {1410.22004},
} -
[KrophollerLinnellMoody88] P. H. Kropholler, P. A. Linnell, and J. A. Moody, "Applications of a new $K$-theoretic theorem to soluble group rings," Proc. Amer. Math. Soc., vol. 104, iss. 3, pp. 675-684, 1988.
@ARTICLE{KrophollerLinnellMoody88,
author = {Kropholler, P. H. and Linnell, P. A. and Moody, J. A.},
title = {Applications of a new {$K$}-theoretic theorem to soluble group rings},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {104},
year = {1988},
number = {3},
pages = {675--684},
issn = {0002-9939},
mrclass = {16A27 (16A08 16A34)},
mrnumber = {0964842},
mrreviewer = {L. N. Vaserstein},
doi = {10.2307/2046771},
url = {https://doi.org/10.2307/2046771},
zblnumber = {0691.16013},
} -
[Lafforgue02] V. Lafforgue, "$K$-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes," Invent. Math., vol. 149, iss. 1, pp. 1-95, 2002.
@ARTICLE{Lafforgue02,
author = {Lafforgue, Vincent},
title = {{$K$}-théorie bivariante pour les algèbres de {B}anach et conjecture de {B}aum-{C}onnes},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {149},
year = {2002},
number = {1},
pages = {1--95},
issn = {0020-9910},
mrclass = {19K35 (46H25 46L80 46M20 58J22)},
mrnumber = {1914617},
mrreviewer = {Georges Skandalis},
doi = {10.1007/s002220200213},
url = {https://doi.org/10.1007/s002220200213},
zblnumber = {1084.19003},
} -
[Linnell93] P. A. Linnell, "Division rings and group von Neumann algebras," Forum Math., vol. 5, iss. 6, pp. 561-576, 1993.
@ARTICLE{Linnell93,
author = {Linnell, Peter A.},
title = {Division rings and group von {N}eumann algebras},
journal = {Forum Math.},
fjournal = {Forum Mathematicum},
volume = {5},
year = {1993},
number = {6},
pages = {561--576},
issn = {0933-7741},
mrclass = {20C07 (16K40 16S35 22D25 46L10)},
mrnumber = {1242889},
mrreviewer = {Alain Valette},
doi = {10.1515/form.1993.5.561},
url = {https://doi.org/10.1515/form.1993.5.561},
zblnumber = {0794.22008},
} -
[Lueck02] W. Lück, $L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory, Springer-Verlag, Berlin, 2002, vol. 44.
@BOOK{Lueck02, key={L{ü}c02},
author = {Lück, Wolfgang},
title = {{$L^2$}-Invariants: Theory and Applications to Geometry and {$K$}-Theory},
series = {Ergeb. Math. Grenzgeb.},
volume = {44},
publisher = {Springer-Verlag, Berlin},
year = {2002},
pages = {xvi+595},
isbn = {3-540-43566-2},
mrclass = {58J22 (19K56 46L80 57Q10 57R20 58J52)},
mrnumber = {1926649},
mrreviewer = {Thomas Schick},
doi = {10.1007/978-3-662-04687-6},
url = {https://doi.org/10.1007/978-3-662-04687-6},
zblnumber = {1009.55001},
} -
[MineyevYu02] I. Mineyev and G. Yu, "The Baum-Connes conjecture for hyperbolic groups," Invent. Math., vol. 149, iss. 1, pp. 97-122, 2002.
@ARTICLE{MineyevYu02,
author = {Mineyev, Igor and Yu, Guoliang},
title = {The {B}aum-{C}onnes conjecture for hyperbolic groups},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {149},
year = {2002},
number = {1},
pages = {97--122},
issn = {0020-9910},
mrclass = {20F67 (19K56 20F65 22D25 46L89)},
mrnumber = {1914618},
mrreviewer = {Alain Valette},
doi = {10.1007/s002220200214},
url = {https://doi.org/10.1007/s002220200214},
zblnumber = {1038.20030},
} -
[Mirowicz91] M. Mirowicz, "Units in group rings of the infinite dihedral group," Canad. Math. Bull., vol. 34, iss. 1, pp. 83-89, 1991.
@ARTICLE{Mirowicz91,
author = {Mirowicz, Maciej},
title = {Units in group rings of the infinite dihedral group},
journal = {Canad. Math. Bull.},
fjournal = {Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques},
volume = {34},
year = {1991},
number = {1},
pages = {83--89},
issn = {0008-4395},
mrclass = {16U60},
mrnumber = {1108933},
mrreviewer = {Sudarshan K. Sehgal},
doi = {10.4153/CMB-1991-013-4},
url = {https://doi.org/10.4153/CMB-1991-013-4},
zblnumber = {0737.16021},
} -
[Murray21] A. G. Murray, More counterexamples to the unit conjecture for group rings, 2021.
@MISC{Murray21,
author = {Murray, A. G},
title = {More counterexamples to the unit conjecture for group rings},
year = {2021},
arxiv = {2106.02147},
zblnumber = {},
} -
[Passman85] D. S. Passman, The Algebraic Structure of Group Rings, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985.
@BOOK{Passman85,
author = {Passman, Donald S.},
title = {The Algebraic Structure of Group Rings},
note = {reprint of the 1977 original},
publisher = {Robert E. Krieger Publishing Co., Inc., Melbourne, FL},
year = {1985},
pages = {xiv+734},
isbn = {0-89874-789-9},
mrclass = {16-02 (16A26 16A27)},
mrnumber = {0798076},
zblnumber = {0654.16001},
} -
[Promislow88] D. S. Promislow, "A simple example of a torsion-free, nonunique product group," Bull. London Math. Soc., vol. 20, iss. 4, pp. 302-304, 1988.
@ARTICLE{Promislow88,
author = {Promislow, S. David},
title = {A simple example of a torsion-free, nonunique product group},
journal = {Bull. London Math. Soc.},
fjournal = {The Bulletin of the London Mathematical Society},
volume = {20},
year = {1988},
number = {4},
pages = {302--304},
issn = {0024-6093},
mrclass = {20F05 (20C07)},
mrnumber = {0940281},
mrreviewer = {Stephen J. Pride},
doi = {10.1112/blms/20.4.302},
url = {https://doi.org/10.1112/blms/20.4.302},
zblnumber = {0662.20022},
} -
[RipsSegev87] E. Rips and Y. Segev, "Torsion-free group without unique product property," J. Algebra, vol. 108, iss. 1, pp. 116-126, 1987.
@ARTICLE{RipsSegev87,
author = {Rips, Eliyahu and Segev, Yoav},
title = {Torsion-free group without unique product property},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {108},
year = {1987},
number = {1},
pages = {116--126},
issn = {0021-8693},
mrclass = {20F06 (20C07)},
mrnumber = {0887195},
mrreviewer = {Stephen J. Pride},
doi = {10.1016/0021-8693(87)90125-6},
url = {https://doi.org/10.1016/0021-8693(87)90125-6},
zblnumber = {0614.20021},
} -
[Sandling81] R. Sandling, "Graham Higman’s thesis “Units in group rings”," in Integral Representations and Applications, Springer, Berlin-New York, 1981, vol. 882, pp. 93-116.
@INCOLLECTION{Sandling81,
author = {Sandling, Robert},
title = {Graham {H}igman's thesis ``{U}nits in group rings''},
booktitle = {Integral Representations and Applications},
venue = {{O}berwolfach, 1980},
series = {Lecture Notes in Math.},
volume = {882},
pages = {93--116},
publisher = {Springer, Berlin-New York},
year = {1981},
mrclass = {20C05 (01A60 01A70 16A25 16A26 16A27)},
mrnumber = {0646094},
mrreviewer = {B. H. Neumann},
doi = {10.1007/BFb0092488},
url = {https://doi.org/10.1007/BFb0092488},
zblnumber = {0468.16013},
} -
[Soelberg18] L. J. Soelberg, Finding torsion-free groups which do not have the unique product property, 2018.
@MISC{Soelberg18,
author = {Soelberg, L. J.},
title = {Finding torsion-free groups which do not have the unique product property},
note = {Master's thesis, Brigham Young University},
year = {2018},
url = {https://scholarsarchive.byu.edu/etd/6932},
zblnumber = {},
} -
[Steenbock15] M. Steenbock, "Rips-Segev torsion-free groups without the unique product property," J. Algebra, vol. 438, pp. 337-378, 2015.
@ARTICLE{Steenbock15,
author = {Steenbock, Markus},
title = {Rips-{S}egev torsion-free groups without the unique product property},
journal = {J. Algebra},
fjournal = {Journal of Algebra},
volume = {438},
year = {2015},
pages = {337--378},
issn = {0021-8693},
mrclass = {20F06 (20F60 20F67 20P05)},
mrnumber = {3353035},
mrreviewer = {Arye Juh\'{a}sz},
doi = {10.1016/j.jalgebra.2015.05.004},
url = {https://doi.org/10.1016/j.jalgebra.2015.05.004},
zblnumber = {1402.20046},
} -
[Strojnowski80] A. Strojnowski, "A note on u.p. groups," Comm. Algebra, vol. 8, iss. 3, pp. 231-234, 1980.
@ARTICLE{Strojnowski80,
author = {Strojnowski, Andrzej},
title = {A note on u.p. groups},
journal = {Comm. Algebra},
fjournal = {Communications in Algebra},
volume = {8},
year = {1980},
number = {3},
pages = {231--234},
issn = {0092-7872},
mrclass = {20E99},
mrnumber = {0558112},
doi = {10.1080/00927878008822456},
url = {https://doi.org/10.1080/00927878008822456},
zblnumber = {0423.20005},
} -
[Valette02] A. Valette, Introduction to the Baum-Connes Conjecture, Birkhäuser Verlag, Basel, 2002.
@BOOK{Valette02,
author = {Valette, Alain},
title = {Introduction to the {B}aum-{C}onnes Conjecture},
series = {Lectures in Math. ETH Zürich},
note = {from notes taken by Indira Chatterji, with an appendix by Guido Mislin},
publisher = {Birkhäuser Verlag, Basel},
year = {2002},
pages = {x+104},
isbn = {3-7643-6706-7},
mrclass = {58J22 (19K35 22D25 46L80 46L87)},
mrnumber = {1907596},
mrreviewer = {Paul D. Mitchener},
doi = {10.1007/978-3-0348-8187-6},
url = {https://doi.org/10.1007/978-3-0348-8187-6},
zblnumber = {1136.58013},
}
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