Combinatorica

, Volume 17, Issue 4, pp 555–575

Expanding graphs and invariant means

  • Yehuda Shalom
Article

DOI: 10.1007/BF01195004

Cite this article as:
Shalom, Y. Combinatorica (1997) 17: 555. doi:10.1007/BF01195004

Abstract

All the known explicit constructions of expander families are essentially obtained by considering a sequence of finite index normal subgroupsNi◃Γ, and taking the Cayley graphs of Γ/Ni w.r. to the projection of aglobal finite set of generators of Γ. For many of these examples (e.g. Γ=SL2ℤ, Γ/NiSL2(\(\mathbb{F}_p \)) we present first constructions of new, different, sets of generators for the finite quotients, which make the Cayley graphs an expander family. An intrinsic connection between the expanding property and uniqueness of the Haar measure on an appropriate compact group, as an invariant mean, is established and used in the construction of such generators.

Mathematics Subject Classification (1991)

05C25 20F32 22E40 28D15 

Copyright information

© János Bolyai Mathematical Society 1997

Authors and Affiliations

  • Yehuda Shalom
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael

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