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Expanding graphs and invariant means

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Abstract

All the known explicit constructions of expander families are essentially obtained by considering a sequence of finite index normal subgroupsN i ◃Γ, and taking the Cayley graphs of Γ/N i w.r. to the projection of aglobal finite set of generators of Γ. For many of these examples (e.g. Γ=SL 2ℤ, Γ/N i ≅SL 2(\(\mathbb{F}_p \)) 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.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University, 91904, Jerusalem, Israel

    Yehuda Shalom

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  1. Yehuda Shalom
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Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).

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Shalom, Y. Expanding graphs and invariant means. Combinatorica 17, 555–575 (1997). https://doi.org/10.1007/BF01195004

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