Let $X$ be a metric space. We say that $X$ is *asymptotically geodesic* if for all $epsilon > 0$, there exists $R > 0$ such that, for all $x,y in X$, there exists some finite sequence of points $p_0=x,p_1,p_2,…,p_N = y in X$ such that each $d(p_{i-1},p_i) le R$ and

$$ sum_{i=1}^N d(p_{i-1},p_i) le (1 + epsilon)d(x,y). $$

I came across this definition while studying finitely generated nilpotent groups and their scaling limits. Pansu proved in “Croissance des boules et des geodesiques fermees dans les nilvarietes” that, if a finitely generated (virtually) nilpotent group $Gamma$ is endowed with a left-invariant metric $d$ such that $(Gamma,d)$ is asymptotically geodesic, then the sequence of metric spaces $(Gamma,frac{1}{n}d)$ has a Gromov-Hausdorff limit (which he describes).

However, I’ve been having a lot of difficulties coming up with examples of metrics on nilpotent groups which *aren’t* asymptotically geodesic, particularly under the extra assumption that the metric in question is bi-Lipschitz to a word metric on $Gamma$.

I believe that it follows pretty quickly from Fekete’s lemma that any invariant metric on $mathbb{Z}$ which is bi-Lipschitz to the standard metric is asymptotically geodesic, and, unless I made a mistake, the same is true for $mathbb{Z}^d$ by a more involved but fairly similar argument. But the property of being asymptotically geodesic is certainly not preserved by bi-Lipschitz maps *a priori*, and it seems implicit in the literature that there should be some metrics bi-Lipschitz to word metrics which are not asymptotically geodesic. (For instance, in Benjamini and Tessera, “First passage percolation in nilpotent Cayley graphs and beyond”, one of the key ingredients in a proof is showing that a certain invariant metric is asymptotically geodesic, even though that metric is assumed to be bi-Lipschitz to a word metric).

So my question is:

(1) What is an example of a left-invariant metric on a finitely generated nilpotent group which is bi-Lipschitz to a word metric but not asymptotically geodesic?

As it happens, I actually have had difficulty constructing such a metric even if I drop the assumption that it is bi-Lipschitz to a word metric, so I would also be interested in the weaker question:

(2) What is an example of a left-invariant metric on a finitely generated nilpotent group which is not asymptotically geodesic?

And even though I’m interested primarily in the case of nilpotent groups, I’m also generally trying to understand this property, so an answer to the even weaker question might help too:

(3) What is an example of a left-invariant metric on a (finitely generated?) discrete group which is not asymptotically geodesic?