On sofic approximations of Property (T) groups

Constraint metric approximations and equations in groups

We introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation. This is done in the language of group equations with coefficients. We give an example of a group which is not constraintly sofic. In building it, we find a sofic representation of free group with trivial commutant among extreme points of the convex structure on the space of sofic representations. We consider the centralizer equation in permutations as an instance of this new general setting. We characterize permutations $p\in S_k$ whose centralizer is stable in permutations with r…

Aldous–Lyons 猜想 I：子群检验

The Aldous--Lyons Conjecture I: Subgroup Tests

本文通过引入子群检验框架，结合非局部博弈与 MIP*=RE 归约，证明了自由群中存在非 co-sofic 不变随机子群，从而否定解决 Aldous–Lyons 猜想。

Combinatorial cost: a coarse setting

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 n…

The topological strong spatial mixing property and new conditions for pressure approximation

In the context of stationary $\mathbb{Z}^d$ nearest-neighbour Gibbs measures $μ$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property (TSSM)) on the support of $μ$ sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of TSSM for studying strong spatial mixing on systems with hard constraints. We also show that TSSM is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by D. Gamarnik and D…