The combinatorial cost

TL;DR

This paper introduces combinatorial invariants called cost and β-invariants for sequences of finite graphs, relating them to existing invariants in group theory and measurable equivalence relations.

Contribution

It defines new combinatorial invariants and explores their relationships with known group invariants like rank gradient and homology gradient.

Findings

Cost and β-invariants are introduced for graph sequences.

Relationships between these invariants and group invariants are established.

The invariants provide new insights into the structure of finite graph sequences.

Abstract

We study the combinatorial analogues of the classical invariants of measurable equivalence relations. We introduce the notion of cost and $\beta$-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.