# Finitely generated group quasi-isometric to finitely presented group

Let $$G$$ be finitely presented and let $$H$$ be finitely generated such that $$G$$ is quasi-isometric to $$H$$. Show that $$H$$ is finitely presented.

I am not 100% certain about the definition of quasi-isometry for groups, I believe we can take it to mean that their Cayley graphs are quasi-isometric (which I'm fine with, since I know how to show that if $$S_1$$, $$S_2$$ two finite generating sets of $$G$$ then $$\Gamma(S_1 , G)$$ is quasi-isometric to $$\Gamma(S_2, G)$$).

I started by writing $$G$$ as $$\langle S \mid R\rangle$$, and letting $$T$$ be a finite generating set of $$H$$. Let $$f\colon G\to H$$ be a $$(K,A)$$-quasi-isometry. I note that any $$h\in H$$ is at most a distance $$L$$ from $$f(v)$$ for some $$v\in G$$ ($$L$$ a positive constant independent of $$h$$).

I'm not sure how to proceed.

• What book are you reading? The problem you are trying to solve is a standard but nontrivial result. Commented May 9, 2023 at 21:24
• If you are not even sure about definitions, this problem will be way too hard for you. My suggestion is to focus on easier staff. Commented May 10, 2023 at 16:06
• I am working from lecture notes on a course in geometric group theory at my college. I have taken the prerequisite courses and am happy with the content before this exercise. I know what is meant by a quasi-isometry, net, geodesic metric space, I can prove Milnor-Svarc lemma. I can prove that if $G$ finitely generated and $S_1$, $S_2$ finite generating sets then $\Gamma(S_1, G)$ quasi-isometric to $\Gamma(S_2, G)$; that $G$ is quasi-isometric to finite index $H\leq G$; and that $G$ is quasi-isometric to $G/N$ for finite normal $N$. Commented May 10, 2023 at 19:22
• Then, likely, you are not ready for this problem. Commented May 10, 2023 at 19:24
• I know what is meant by quasi-isometry for metric spaces, and I can associate to each $G$ a finitely generated group and $S\subseteq G$ a finite generating set two metric spaces: the Cayley graph $\Gamma (S,G)$, and the subspace $G$ of vertices. The lecture notes do not specify which of these two should be taken when defining quasi-isometry for groups. Commented May 10, 2023 at 19:25

Alternatively, read Theorem 18.2.12 (for groups of type $$F_2$$) in