Here is a curious question posed to me by Apoorva Khare that I do not know the answer to. Let {F_2} be the free group on two generators {a,b}. Does there exist a metric {d} on this group which is

  • bi-invariant, thus {d(xg,yg)=d(gx,gy) = d(x,y)} for all {x,y,g \in F_2}; and
  • linear growth in the sense that {d(x^n,1) = n d(x,1)} for all {x \in F_2} and all natural numbers {n}?

By defining the “norm” of an element {x \in F_2} to be {\| x\| := d(x,1)}, an equivalent formulation of the problem asks if there exists a non-negative norm function {\| \|: F_2 \rightarrow {\bf R}^+} that obeys the conjugation invariance

\displaystyle  \| gxg^{-1} \| = \|x \| \ \ \ \ \ (1)

for all {x,g \in F_2}, the triangle inequality

\displaystyle  \| xy \| \leq \| x\| + \| y\| \ \ \ \ \ (2)

for all {x,y \in F_2}, and the linear growth

\displaystyle  \| x^n \| = |n| \|x\| \ \ \ \ \ (3)

for all {x \in F_2} and {n \in {\bf Z}}, and such that {\|x\| > 0} for all non-identity {x \in F_2}. Indeed, if such a norm exists then one can just take {d(x,y) := \| x y^{-1} \|} to give the desired metric.

One can normalise the norm of the generators to be at most {1}, thus

\displaystyle  \| a \|, \| b \| \leq 1.

This can then be used to upper bound the norm of other words in {F_2}. For instance, from (1), (3) one has

\displaystyle  \| aba^{-1} \|, \| b^{-1} a b \|, \| a^{-1} b^{-1} a \|, \| bab^{-1}\| \leq 1.

A bit less trivially, from (3), (2), (1) one can bound commutators as

\displaystyle  \| aba^{-1} b^{-1} \| = \frac{1}{3} \| (aba^{-1} b^{-1})^3 \|

\displaystyle  = \frac{1}{3} \| (aba^{-1}) (b^{-1} ab) (a^{-1} b^{-1} a) (b ab^{-1}) \|

\displaystyle  \leq \frac{4}{3}.

In a similar spirit one has

\displaystyle  \| aba^{-2} b^{-1} \| = \frac{1}{2} \| (aba^{-2} b^{-1})^2 \|

\displaystyle  = \frac{1}{2} \| (aba^{-1}) (a^{-1} b^{-1} a) (ba^{-1} b^{-1}) (ba^{-1} b^{-1}) \|

\displaystyle  \leq 2.

What is not clear to me is if one can keep arguing like this to continually improve the upper bounds on the norm {\| g\|} of a given non-trivial group element {g} to the point where this norm must in fact vanish, which would demonstrate that no metric with the above properties on {F_2} would exist (and in fact would impose strong constraints on similar metrics existing on other groups as well). It is also tempting to use some ideas from geometric group theory (e.g. asymptotic cones) to try to understand these metrics further, though I wasn’t able to get very far with this approach. Anyway, this feels like a problem that might be somewhat receptive to a more crowdsourced attack, so I am posing it here in case any readers wish to try to make progress on it.