Vertex-transitive graphs with uniformly bisecting quasi-geodesics
Joseph MacManus
TL;DR
This work proves that an infinite, locally finite, quasi-transitive graph X satisfying a uniformly bisecting quasi-geodesics property must be quasi-isometric to either the Euclidean plane $\mathbb{R}^2$ or the hyperbolic plane $\mathbb{H}^2$. The argument splits into non-hyperbolic and hyperbolic cases: non-hyperbolic leads to quadratic growth via Varopoulos-type isoperimetric controls and Trofimov–Bass–Guivarc'h machinery, yielding a virtual $\mathbb{Z}^2$-type structure and Euclidean rigidity; hyperbolic yields a circle boundary and, by the convergence group theorem, virtual Fuchsianity corresponding to $\mathbb{H}^2$. In particular, for Cayley graphs, the underlying group is a virtual surface group, providing a geometric characterization that extends known results about hyperbolic groups with circular boundary. The paper also resolves Kourov notebook Problem 14.98 in the affirmative and develops a suite of geometric tools—witnesses, cross-examiners, and jurisdiction criteria—that link coarse separation properties to large-scale growth and rigidity.
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.