G. Elek defined hyperfiniteness for graph sequences in 2007. He proved that, for G a residually finite group, amenability of G is equivalent to hyperfiniteness of some box space. This must be compared to a result by J. Roe (2005): amenability of G is equivalent to Yu's property A for some box space.
In this talk I will report on recent results by my student T. Kaiser (2017). He noticed that the above characterizations of amenability still hold true when a box space is replaced by a sequence of Schreier graphs associated with a Farber sequence; but it stops being true for more general sofic approximations of G. Indeed, V. Alekseev and M. Finn-Sell proved in 2016 that if a sofic approximation of G has property A, then G is amenable, but the converse is false. This led Kaiser to introduce property almost-A, to re-establish an equivalence in the above results.
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