Asymptotic half-grid and full-grid minors

TL;DR

This work resolves part of Georgakopoulos and Papasoglu’s coarse-grid problem by showing that locally finite, quasi-transitive graphs with a thick end and cycle space generated by bounded-length cycles contain the full-grid $FG$ as an asymptotic minor and as a diverging minor; a parallel half-grid result holds for graphs of finite maximum degree. The authors develop new coarse-geometry tools—ultra-fat minors and escaping subdivisions—to convert thick-end structures into fat/diverging grid-minors, with a coherent pathway from half-grid to full-grid results. The methods yield applications to locally finite Cayley graphs of finitely presented groups, and they tie into broader themes in coarse graph theory, including coarse embeddings and quasi-isometries to trees or planar graphs. Overall, the paper advances understanding of when grid-like asymptotic structures must appear in coarse graphs and how they relate to transitivity, ends, and the cycle space, offering partial resolutions to open problems and guiding future generalizations beyond finitely presented groups.

Abstract

We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of finitely presented groups that are not virtually free, and partially solves problems of Georgakopoulos and Papasoglu and of Georgakopoulos and Hamann. Additionally, we show that every (not necessarily quasi-transitive) graph of finite maximum degree which has a thick end and whose cycle space is generated by cycles of bounded length contains the half-grid as an asymptotic minor and as a diverging minor.

Figures (8)

• Figure 2.1: The hexagonal full-grid with vertical double rays $R^i$ and horizontal edges $e_{ij}$.
• Figure 3.1: The hexagonal half-grid (full-grid) with an enumeration of its horizontal edges as needed for the proof of \ref{['prop:HexGridAfterDeletingPaths']}.
• Figure 3.2: Depicted in blue and green are the subgraphs $X_{32}$ and $Y_{32}$ that are used to choose $Q_{32}$. The paths $Q_{ij}$ that are chosen before $Q_{32}$ are shown in grey.
• Figure 3.3: The black subdivision $\widetilde{H}$ of the hexagonal half-grid is a subgraph of the grey subdivision $H'$ of the hexagonal half-grid.
• Figure 6.1: Sketch of a $K_5$ minor in a half-grid with all $(i,0)(i+1,1)$ and $(i,1)(i+1,0)$ edges. The branch paths $E_{ij}$ between $V_5$ and the $V_i$'s are the thickened $(i,0)(i+1,0)$ edges.

Theorems & Definitions (73)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1
• Theorem 2.2
• Proposition 3.2
• proof
• Theorem 3.3
• Theorem 3.4