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The combinatorial cost

Gábor Elek
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Abstract

We study the combinatorial analogues of the classical invariants of measurable equivalence relations. We introduce the notion of cost and $β$-invariants (the analogue of the first $L^2$-Betti number introduced by Gaboriau) for sequences of finite graphs with uniformly bounded vertex degrees and examine the relation of these invariants and the rank gradient resp. mod $p$ homology gradient invariants introduced by Lackenby for residually finite groups.

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Gábor Elek. 2006-08-18. The combinatorial cost. https://arxiv.org/abs/math/0608474

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