We prove that in a certain statistical sense the Cayley graph of almost every finitely presented group with $m\ge 2$ generators contains a subdivision of the complete graph on $l\le 2m+1$ vertices. In particular, this Cayley graph is non planar. We also show that some group constructions preserve the planarity.
"On the Cayley graph of a generic finitely presented group." Bull. Belg. Math. Soc. Simon Stevin 11 (4) 589 - 601, November 2004. https://doi.org/10.36045/bbms/1102689123