Almost planar finitely presented groups

TL;DR

The paper proves that finitely presented groups with k-planar Cayley graphs have finite-index subgroups with planar Cayley graphs, and shows that certain coarsely simply connected, k-planar graphs are quasi-isometric to planar graphs.

Contribution

It establishes a connection between k-planar Cayley graphs and the existence of planar subgroups, and characterizes certain coarsely simply connected graphs as quasi-isometric to planar graphs.

Findings

Finitely presented groups with k-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs.

k-planar, coarsely simply connected graphs are quasi-isometric to planar graphs.

Abstract

We show that finitely presented groups which admit $k$-planar Cayley graphs contain finite-index subgroups with planar Cayley graphs. More generally, we answer a question of Georgakopoulos and Papasoglu in the special case of coarsely simply connected graphs: a $k$-planar, coarsely simply connected, connected, locally finite, quasi-transitive graph is quasi-isometric to a planar graph.