Combinatorial cost: A coarse setting

Abstract

The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 and hyperfiniteness are coarse invariants. We also show 'cost−1' for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A.

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

“…For the case where M = Hilbert, V. Alekseev and Finn-Sell [AFS16] extended the framework of Theorem A for the case where (G m ) m is a LEF approximation of G ∞ , see Definition 3.2, to a sofic approximation of a sofic group. However, in that generality, only one direction (the direction of (i) in Theorem A) can be deduced; see the construction of a counterexample to the other direction by T. Kaiser [Kai17], which is explained below Theorem 5.3 in the concerning reference [Kai17]. Compare also with our points (a), (b) with LEA approximations, and the case where M is general.…”

“…The two authors thank Damian Osajda for discussion on his construction of RF groups without property A, specially for Remark 9.5. They are grateful to Sylvain Arnt for discussion and the terminology of a-M-menability, Goulnara Arzhantseva and Florent Baudier for discussion and comments, Tom Kaiser for the reference [Kai17], and Hiroyasu Izeki, Shin Nayatani and Tetsu Toyoda for discussions on r-uniformly convex metric spaces and the Izeki-Nayatani invariants. The first-named author thanks Yash Lodha for several comments, including advice to build Section 8 as a distinct section, and his suggestion of the terminology of "fragmentary actions", and Manor Mendel for discussion on several concepts on metric geometry of non-linear spaces.…”

Uniform local amenability implies Property A

“…For a finite index subgroup H, we can consider the left Schreier graph Sch(Γ/H, Σ). Kaiser [9] proved that if Γ is a finitely generated group and Γ ⊃ H 1 ⊃ H 2 ⊃ . .…”

Non-standard limits of graphs and some orbit equivalence invariants

“…We briefly mention an extension we do not consider in this version in order to limit the length of the present article. There exists also a notion of coarse equivalence between pairs of sequences (G n , Φ n ) n and (H n , Ψ n ) n of graphed groupoids (compare [AFS19] and [Kai19] which considers finite structures). The groupoids G n and H n are in particular stably orbit equivalent, so that they are related by a compression constant ι(G n , H n ).…”