Is there any known example of a one-ended finitely presented group with exponential growth that does not contain a quasi-isometric copy of the hyperbolic plane?


This question is motivated by the following question of Papasoglu mentioned in the paper *'Quasi-hyperbolic planes in hyperbolic group'* by Bonk–Kleiner which asks whether every one-ended finitely presented group $G$ contains a quasi plane- the image of a uniform embedding $P\rightarrow G$ where $P$ is a complete Riemannian plane with bounded geometry.


Thank you for reading this.



Groups that don't contain quasi-hyperbolic plane
It is a result of Buyalo and Schroeder [BS, Corollary 1.2] that for every $n\ge 2$ there is no QI embedding of the hyperbolic space $\mathbb{H}^n\_\mathbf{R}$ into any product of $n-1$ trees with a Euclidean space.


In particular, the 1-ended group $F\_2\times\mathbf{Z}$ (as well as $F\_2\times\mathbf{Z}^d$ for arbitrary $d\ge 1$) contains no QI-copy of the hyperbolic plane, thus answering your question.


[BS] S. Buyalo and V. Schroeder. The hyperbolic dimension of metric spaces. Algebra i Analiz, 19(1):93–108, 2007. [ArXiv link](https://arxiv.org/abs/math/0404525)


