Startseite Free by cyclic groups and linear groups with restricted unipotent elements
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Free by cyclic groups and linear groups with restricted unipotent elements

  • Jack O. Button EMAIL logo
Veröffentlicht/Copyright: 24. Oktober 2017
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

We introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic. We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6]. We give examples of abstract groups lying in this class, but also show that Gersten’s free by cyclic group does not. This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group.

MSC 2010: 20F67; 20F65

References

[1] M. Aschenbrenner, S. Friedl and H. Wilton, 3-Manifold Groups, EMS Ser. Lect. Math., European Mathematical Society, Zürich, 2015. 10.4171/154Suche in Google Scholar

[2] V. G. Bardakov and O. V. Bryukhanov, On linear representations of some extensions (in Russian), Vestnik Novosib. Univ. Ser. Mat. Mekh. Inf. 7 (2007), 45–58. Suche in Google Scholar

[3] S. J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471–486. 10.1090/S0894-0347-00-00361-1Suche in Google Scholar

[4] J. O. Button, A 3-manifold group which is not four dimensional linear, J. Pure Appl. Algebra 218 (2014), no. 9, 1604–1619. 10.1016/j.jpaa.2014.01.001Suche in Google Scholar

[5] J. O. Button, Minimal dimension faithful linear representations of common finitely presented groups, preprint (2016), https://arxiv.org/abs/1610.03712. Suche in Google Scholar

[6] J. O. Button, Properties of non positive curvature for linear groups with restricted unipotent elements, submitted. Suche in Google Scholar

[7] B. Fine and G. Rosenberger, A note on faithful representations of limit groups, Groups Complex. Cryptol. 3 (2011), no. 2, 349–355. 10.1515/gcc.2011.014Suche in Google Scholar

[8] B. Fine and G. Rosenberger, Faithful representations of limit groups II, Groups Complex. Cryptol. 5 (2013), no. 1, 91–96. 10.1515/gcc-2013-0005Suche in Google Scholar

[9] E. Formanek and C. Procesi, The automorphism group of a free group is not linear, J. Algebra 149 (1992), no. 2, 494–499. 10.1016/0021-8693(92)90029-LSuche in Google Scholar

[10] S. M. Gersten, The automorphism group of a free group is not a CAT(0) group, Proc. Amer. Math. Soc. 121 (1994), no. 4, 999–1002. Suche in Google Scholar

[11] M. F. Hagen and D. T. Wise, Cubulating hyperbolic free-by-cyclic groups: The general case, Geom. Funct. Anal. 25 (2015), no. 1, 134–179. 10.1007/s00039-015-0314-ySuche in Google Scholar

[12] T. Hsu and D. T. Wise, On linear and residual properties of graph products, Michigan Math. J. 46 (1999), no. 2, 251–259. 10.1307/mmj/1030132408Suche in Google Scholar

[13] S.-H. Kim and T. Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013), no. 1, 493–530. 10.2140/gt.2013.17.493Suche in Google Scholar

[14] D. Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131–156. 10.2307/3062152Suche in Google Scholar

[15] A. Lubotzky and D. Segal, Subgroup Growth, Progr. Math. 212, Birkhäuser, Basel, 2003. 10.1007/978-3-0348-8965-0Suche in Google Scholar

[16] V. D. Mazurov and E. I. Khukhro, The Kourovka Notebook. Unsolved Problems in Group Theory, 18th ed., Russian Academy of Sciences Siberian Division, Novosibirsk, 2014. Suche in Google Scholar

[17] G. A. Niblo and D. T. Wise, Subgroup separability, knot groups and graph manifolds, Proc. Amer. Math. Soc. 129 (2001), no. 3, 685–693. 10.1090/S0002-9939-00-05574-XSuche in Google Scholar

[18] V. L. Nisnewitsch, Über Gruppen, die durch Matrizen über einem kommutativen Feld isomorph darstellbar sind, Rec. Math. N.S. 8 (50) (1940), 395–403. Suche in Google Scholar

[19] V. A. Roman’kov, The linearity problem for the unitriangular automorphism groups of free groups, J. Siberian Federal Univ. Math. Phys. 6 (2013), 516–520. Suche in Google Scholar

[20] P. B. Shalen, Linear representations of certain amalgamated products, J. Pure Appl. Algebra 15 (1979), no. 2, 187–197. 10.1016/0022-4049(79)90033-1Suche in Google Scholar

[21] B. A. F. Wehrfritz, Generalized free products of linear groups, Proc. Lond. Math. Soc. (3) 27 (1973), 402–424. 10.1112/plms/s3-27.3.402Suche in Google Scholar

[22] B. A. F. Wehrfritz, Infinite Linear Groups, Ergeb. Math. Grenzgeb. (3) 76, Springer, New York, 1973. 10.1007/978-3-642-87081-1Suche in Google Scholar

Received: 2017-5-3
Published Online: 2017-10-24
Published in Print: 2017-11-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 5.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/gcc-2017-0009/html?lang=de
Button zum nach oben scrollen