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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.

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    $\begingroup$ I don't know. It's true for polycyclic groups and Baumslag-Solitar groups $\mathrm{BS}(m,n)$ for $0<m<n$. However I'm not sure for $\mathbf{F}_2\times\mathbf{Z}$. $\endgroup$
    – YCor
    Commented Feb 9, 2023 at 16:03
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    $\begingroup$ Thank you so much. Can you please give me a reference or any insights about why it is true for any polycyclic group with exponential growth? $\endgroup$
    – user429294
    Commented Feb 9, 2023 at 18:18
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    $\begingroup$ For polycyclic groups I just checked it, I don't think it's written. A polycyclic group of exp. growth is QI to some real triangulable Lie group. And the latter contains a QI-embedded 2-dimensional affine group $\mathbf{R}\rtimes\mathbf{R}$. $\endgroup$
    – YCor
    Commented Feb 9, 2023 at 19:31
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    $\begingroup$ And I tend to believe that $F_2\times\mathbf{Z}$ has no QI-embedded $\mathbf{H}^2$. Intuitively the QI-embedded planes therein should be Euclidean. $\endgroup$
    – YCor
    Commented Feb 9, 2023 at 19:32
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    $\begingroup$ Maybe this is trivial but can I ask why that QI-embedded 2-dimensional affine group obstruct QI-embedding of the hyperbolic plane? $\endgroup$
    – user429294
    Commented Feb 9, 2023 at 21:07

1 Answer 1

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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

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