Local-to-global covering rigidity for Cayley graphs

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finitely generated group $\Gamma$ and finite symmetric generating set $S$ not containing the identity, its Cayley graph has vertices $\Gamma$ and edges $\{g,gs\}$ for $s\in S$. For a graph $H$ with unit edges, $B_H(v,r)$ is the induced rooted graph on vertices of distance at most $r$ from $v$. A graph covering $q:G\to H$ is a surjective graph map inducing a bijection from the neighbors of each vertex $v$ onto the neighbors of $q(v)$.

Statement recorded in the supplied source.

For every Cayley graph $G$, is there an integer $r=r(G)\ge1$ such that every connected graph $H$ whose rooted radius-$r$ balls are isomorphic to $B_G(1,r)$ admits a covering map $G\to H$? [S2]

Short English statement

Does sufficiently large local agreement with a Cayley graph force a global graph covering from it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000016 (AMR-099-0016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Coarse Geometry and Randomness, Saint-Flour lecture notes, version dated 30 October 2013; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 2, Open Problem 2.4, p. 22. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.