On quasi-transitive amenable graphs

Quasi-vertex-transitive Maps on the Plane

Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of type $[5^3, 3]$.

关于拟等距于平面（Cayley）图的拟传递图的一个注记

A note on quasi-transitive graphs quasi-isometric to planar (Cayley) graphs

本文证明拟等距于平面图的拟传递图可升级为平面 Cayley 图，回答了 Esperet--Giocanti 和 Hamann 的开放问题。

Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$κ$-to-one in a natural sense for some $κ>0$. For non-amenable graphs, all quasi-isometries are quasi-$κ$-to-one for any $κ>0$, while for amenable ones there exists at most one possible such $κ$. For an amenable graph $X$, we show that the set of possible $κ$ forms a subgroup of $\mathbb{R}_{>0}$ that we call the (measure-)scaling group of $X$. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a giv…

Topological description of Riemannian foliations with dense leaves

A transitive compact foliated space is shown to be a Riemannian foliation if and only if it is locally connected, finite dimensional, strongly equicontinuous and quasi-analytic, and the closure of its holonomy pseudogroup is quasi-analytic.

An Inaccessible Graph

An inaccessible, vertex transitive, locally finite graph is described. This graph is not quasi-isometric to a Cayley graph.