Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-29T18:54:50.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups of p-deficiency one

Published online by Cambridge University Press:  20 September 2013

ANITHA THILLAISUNDARAM
ANITHA THILLAISUNDARAM*
Affiliation:
Institut für Algebra und Geometrie, Mathematische Fakultät, Otto-von-Guericke-Universität Magdeburg, 39106 Magdeburg, Germany. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 1 , January 2014 , pp. 115 - 121
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Allcock, D.Spotting infinite groups, Math. Proc. Cambridge Phil. Soc. (1) 125 (1999), 39–42.CrossRefGoogle Scholar
[2]Barnea, Y. and Schlage–Puchta, J.-C. On p-deficiency in groups, accepted by J. Group Theory.Google Scholar
[3]Bekka, B., de la Harpe, P. and Valette, A.Kazhdan's Property (T). New Math. Monogr. 11 (Cambridge University Press, 2008).CrossRefGoogle Scholar
[4]Button, J. O.Large groups of deficiency 1. Israel J. Math 167 (2008), 111–140.CrossRefGoogle Scholar
[5]Button, J. O. and Thillaisundaram, A.Applications of p-deficiency and p-largeness. Internat. J. of Algebra Comput. (4) 21 (2011), 547–574.CrossRefGoogle Scholar
[6]Ershov, M.Golod–Shafarevich groups with property (T) and Kac–Moody groups. Duke Math. J. (2) 145 (2008), 309–339.CrossRefGoogle Scholar
[7]Ershov, M. and Jaikin–Zapirain, A.Property (T) for noncommutative universal lattices. Invent. Math. (2) 179 (2010), 303–347.CrossRefGoogle Scholar
[8]Khukhro, E. I. and Mazurov, V. D.Unsolved problems in group theory: the Kourovka notebook. (Russian Academy of Sciences, 1995).Google Scholar
[9]Lackenby, M.Adding high powered relations to large groups. Math. Res. Lett. (6) 14 (2007), 983–993.CrossRefGoogle Scholar
[10]Lackenby, M.Detecting large groups. J. Algebra (10) 324 (2010), 2636–2657.CrossRefGoogle Scholar
[11]Olshanskii, A. Yu. and Osin, D. V.Large groups and their periodic quotients. Proc. Amer. Math. Soc. (3) 136 (2008), 753–759.CrossRefGoogle Scholar
[12]Schlage–Puchta, J.-C. Profinite Groups Meeting, 12th June 2009.Google Scholar
[13]Schlage–Puchta, J.-C.A p-group with positive rank gradient. J. Group Theory (2) 15 (2012), 261–270.CrossRefGoogle Scholar