Example of groups that are not quasi-isometric but have the same growth rate?

Question

I have started working on group growth earlier this year, mainly using Drutu and Kapovich's notes. This morning I found myself wondering if I could find an example of groups that are not quasi-isometric but have the same growth rate. Spontaneously, I thought about finite groups, groups of linear growth and free groups. All those cannot provide such an example.

I firmly believe this to be possible but I have been unable to find one. Google searches have not helped me either so maybe some of you can.

Answer 1

This is not at all an area I know about, but some quick googling gives the following:

• The fundamental group of any closed hyperbolic $n$-manifold is quasi-isometric to $\mathbb{H}^n$, and any two such fundamental groups (for $n \ge 2$) have exponential growth rate.
• $\mathbb{H}^n$ and $\mathbb{H}^m$ are not quasi-isometric if $n \neq m$; they can be distinguished by their Gromov boundaries, which are $S^{n-1}$ and $S^{m-1}$ respectively.

So we can take the fundamental groups of a closed hyperbolic $n$-manifold and $m$-manifold for $n \neq m \ge 2$.

Answer 2

To augment the answer of @QiaochuYuan, I'll turn my comment into an answer: In the realm of groups of polynomial growth, the simplest example is that both $\mathbb Z^4$ and the integer Heisenberg group have polynomial growth of degree $4$.

One still has to prove that those groups are not quasi-isometric to each other. The simplest proof that I know is that $\mathbb Z^4$ acts freely, properly and cocompactly on the topological space $\mathbb R^4$ hence its cohomological dimension is $4$; whereas the integer Heisenberg group acts freely, properly and cocompactly on $\mathbb R^3$ hence its cohomological dimension is $3$. And in the realm of groups that act freely, properly and cocompactly on Euclidean spaces (or even on contractible CW complexes), the cohomological dimension is a quasi-isometry invariant.