Minor-excluded graphs and soficity

TL;DR

The paper proves that unimodular random rooted graphs that are almost surely one-ended and do not contain a fixed finite graph $H$ as a minor are sofic, and that when the graphs are also quasi-transitive and minor-excluded, the graphs are treeable and hence sofic. The core method combines cutting non-expanding regions, Voronoi-based filaments, and a thinning process to achieve locally-small treewidth, after which grid-minor theory yields bounded local treewidth and allows Benjamini–Schramm limits to be approximated by treeable graphs. Percolation stability arguments ensure that passing to limits preserves soficity, and the approach extends to quasi-transitive and transitive minor-excluded classes through canonical tree decompositions and graphings. The results broaden the scope of classes known to be sofic, linking minor-exclusion, ends, and transitivity to structural decompositions and limit objects with practical implications for understanding random limits and group-theoretic connections.

Abstract

A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Given any finite graph $H$, we prove that every one-ended, unimodular random rooted graph that does not have H as a minor must be sofic. The hypothesis regarding the number of ends can be dropped under the additional assumption that the graph is quasi-transitive.

Figures (10)

• Figure 1: On the left, we have depicted a portion of an infinite, one-ended, expanding, planar graph $G$. Some large but finite tree-like structures, called filaments, have been selected throughout the graph, and are highlighted in red. After removing these filaments from the graph, we obtain a new graph $G_{\operatorname{thin}}$, a portion of which is shown on the right. If the filaments are chosen appropriately, then $G_{\operatorname{thin}}$ will have locally-small treewidth, in the sense that any ball in this graph whose radius is not too large will have small treewidth. In essence, the only structures within $G_{\operatorname{thin}}$ which are stopping the graph from having small treewidth are the macroscopic cycles which enclose at least one of the filaments; one such cycle has been marked with dotted yellow lines. Since $G$ is an expander and the filaments are large, any such cycle must also be large.
• Figure 2: The $k\times \ell$-wall, $W_{k\times\ell}$, is simply the skeleton of a hexagonal grid with $k$ rows and $\ell$ columns. The $7\times 7$ wall is shown above.
• Figure 3: Both images depict $W_i$, which is a subdivision of the $5h\times 5h$-wall (some of its edges may be subdivided, even if it is not shown in the picture). On the left, we see a path of length at most $h$ in $G_{\alpha}$ (red) connecting $u_i$ (green) to the filament $\Psi_i$. This path goes through a vertex $v$ (purple) of $W$ that lies outside $W_i$. On the right, two subsections of this path have been highlighted (turquoise). Both of these sections constitute leaps with an endpoint inside $W_i$. If it were the case that no such section exists, then the path joining $u_i$ to $\Psi_i$ would not be long enough to reach a vertex of $W$ outside $W_i$.
• Figure 4: On the left, the grid $G_{7\times 7}$. On the right, the wall $W_{7\times 7}$.
• Figure 5: A crossed-$7\times 7$-grid.

Theorems & Definitions (43)

• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Conjecture 1.6
• Lemma 3.1: Everything shows at the root, Lemma 2.3, aldous2007processes
• Lemma 3.2: Lemma 2.2, angel2018hyperbolic
• Lemma 3.3: Lemma 3.1, angel2018hyperbolic
• Lemma 3.4: Lemma 3.2, angel2018hyperbolic
• Proposition 3.5
• proof