Vertex-transitive graphs with uniformly bisecting quasi-geodesics

TL;DR

This paper characterizes infinite, locally finite, quasi-transitive graphs with bi-infinite quasi-geodesics that separate the graph into two deep parts, showing they are quasi-isometric to Euclidean or hyperbolic planes, extending known classifications.

Contribution

It proves that such graphs are quasi-isometric to classical geometric models, confirming a long-standing problem and linking graph properties to group theoretic classifications.

Findings

Graphs are quasi-isometric to Euclidean or hyperbolic planes.

Cayley graphs with this property correspond to virtual surface groups.

The result extends the classification of hyperbolic groups with circular boundary.

Abstract

Suppose that $X$ is an infinite, connected, locally finite, quasi-transitive graph with the property that every bi-infinite quasi-geodesic uniformly coarsely separates $X$ into exactly two deep pieces. We show that such an $X$ is quasi-isometric to either the Euclidean plane or the hyperbolic plane. In particular, if $X$ is a Cayley graph of a finitely generated group $G$ with the above property, then $G$ is a virtual surface group. This can be interpreted as an extension of the well-known fact that a hyperbolic group with circular boundary is virtually Fuchsian. Our theorem positively resolves Problem 14.98 of the Kourovka Notebook, posed by V. A. Churkin in 1999. The proof uses an isoperimetric inequality of Varopoulos to show that if such a graph has the above property, then either it is hyperbolic or has quadratic growth.