Skip to main content
SpringerLink
Log in

Expansion of coset graphs of PSL2(Fp)

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let G be a finite group and let H1, H2 < G be two subgroups. In this paper, we are concerned with the bipartite graph whose vertices are G/H1 ∪ G/H2 and a coset g1H1 is connected with another coset g2H2 if and only if g1H1∩g2H2= Ø. The main result of the paper establishes the existence of such graphs with large girth and large spectral gap. Lubotzky, Manning and Wilton use such graphs to construct certain infinite groups of interest in geometric group theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL2(Fp), Annals of Mathematics 167 (2008), 625–642.

    Article  MathSciNet  Google Scholar 

  2. G. Frobenius, Über Gruppencharaktere, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1896), 985–1021.

    Google Scholar 

  3. A. Gamburd, S. Hoory, M. Sliahshahani, A. Shalev and B. Virag, On the girth of random Cayley graphs, Random Structures & Algorithms 35 (2009). 100–117.

    Article  MathSciNet  Google Scholar 

  4. H. A. Helfgott, Growth and generation in SL2(ℤ/pℤ), Annals of Mathematics 167 (2008), 601–623.

    Article  MathSciNet  Google Scholar 

  5. E. Kowalski, Explicit growth and expansion for SL2, International Mathematics Research Notices (2013), 5645–5708.

    Google Scholar 

  6. A. Lubotzky, J. F. Manning and H. Wilton, Generalized triangle groups, expanders, and a problem of Agol and Wise, Commentarii Mathematici Helvetici 94 (2019), 53–66.

    Article  MathSciNet  Google Scholar 

  7. P. Sarnak and X. X. Xue, Bounds for multiplicities of automorphic representations, Duke Mathematical Journal 64 (1991), 207–227.

    Article  MathSciNet  Google Scholar 

  8. M. Suzuki, Group Theory I, Grundlehren der Mathematischen Wissenschaften, Vol. 247, Springer, Berlin-New York, 1982.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Péter P. Varjú

Authors
  1. Péter P. Varjú
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Péter P. Varjú.

Additional information

I gratefully acknowledge the support of the Royal Society.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Varjú, P.P. Expansion of coset graphs of PSL2(Fp). Isr. J. Math. 233, 335–349 (2019). https://doi.org/10.1007/s11856-019-1907-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-019-1907-1

Search

Navigation