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mg.metric geometry – Example of an invariant metric on a nilpotent group which isn’t asymptotically geodesic Answer

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mg.metric geometry – Example of an invariant metric on a nilpotent group which isn’t asymptotically geodesic

This respond appears to basically breathe what @YCor was going for in his feedback, however I’ll give very categorical examples, since on the point he described the respond, there have been some particulars that wanted to breathe checked that weren’t instantly limpid to me. (I await that is an arrogate status to respond my avow query.)

First, with none requirement of being bi-Lipschitz to a phrase metric, @YCor’s first observation suggests a really simple instance on $mathbb{Z}$, that’s,
$d(m,n) = sqrtn-m$. It’s valid to bethink that the sq. root of any metric is a metric, and taking the sq. root is a pleasant route to raze asymptotic geodesicity. Anyone studying this could labor out why asymptotic geodesicity fails on this instance in organize to grasp the following one.

Now, for an instance of a metric which is bi-Lipschitz to a phrase metric however not asymptotically geodesic (urged by @YCor’s second observation).
Take $Gamma$ to breathe the Heisenberg group
$$ Gamma := langle X, Y, Z | [X,Y]=Z, [X,Z]=[Y,Z]=1rangle.$$
It’s not too difficult to display that every component of $Gamma$ can breathe written uniquely
as $X^ok Y^l Z^m$, and so now we have a bijection $X^ok Y^l Z^m leftrightarrow (ok,l,m)$
between $Gamma$ and $mathbb{Z}^3$.

It’s too a benchmark rehearse to display that, for some constants $0<c<C<infty$,
$$ c max(|ok|,|l|,sqrtm) le |(ok,l,m)| le C max(|ok|,|l|,sqrtm), $$
the place $|(ok,l,m)|$ is the size of the shortest phrase in $X$ and $Y$ which is the same as $X^ok Y^l Z^m$. (If you have not accomplished this earlier than, the important thing statement is that $|Z^{m^2}| = |[X^m,Y^m]| = O(m)$. It’s too valid within the evaluation to have the next interpretation of the geometry of the Heisenberg group: a phrase in $X,Y$ represents a path within the Cayley graph of $Gamma$ with respect to the mills $X$ and $Y$, and we are able to have a look at the projection of this path onto the Cayley graph of the abelianization
$Gamma^{ab} cong
langle barrier{X}, barrier{Y} | [bar{X},bar{Y}]=1 rangle cong mathbb{Z}^2$
.
If the phrase is the same as $Z^m$, then the projected path in $mathbb{Z}^2$ is closed and $m$ is the same as the (signed) region enclosed by that path; for instance $[X^m,Y^m]=Z^{m^2}$ attracts out a sq. in $mathbb{Z}^2$ with region $m^2$.)

One then has to bridle that (1) $max(|ok|,|l|,sqrtm)$ is in reality subadditive on $Gamma$, and therefore induces an invariant metric $d(x,y)=|x^{-1}y|$ on $Gamma$ which is bi-Lipschitz to the phrase metric, and (2) the induced metric $d$ shouldn’t be asymptotically geodesic. This ought to breathe undoubted, but when one is worried that they energy have made some errors of their evaluation, point to too that for any $epsilon > 0$,
the duty $max(|ok|,|l|,epsilonsqrtm)$ is too bi-Lipschitz, and the smaller $epsilon$ is, the better it’s to corroborate that this duty is subadditive and that it’s not asymptotically geodesic. Here the heuristic to behold that this metric shouldn’t be asymptotically geodesic is the next: one can’t get an approximate geodesic from the identification to $Z^M$ by touring alongside powers of $Z$ for the identical intuition that the sq. root of the benchmark metric on $mathbb{Z}$ shouldn’t be asymptotically geodesic. The paths which ought to approximate geodesics ought to come from phrases in $X$ and $Y$, however the size of such paths will breathe roughly some ceaseless (unbiased of $epsilon$) occasions $sqrt{M}$, whereas the prescribed distance from the identification to $Z^M$ is
$epsilon sqrt{M}$.

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