Given a Riemannian torus $(T,d)$ with fundamental group $pi_1(T)=langle a,b mid ab=ba rangle$. Denote for any $gamma in pi_1(T)$ the infimum length of all representatives of $gamma$ by $L(gamma)$.

Is there a way to get a general upper bound on $L(ab)$ in terms of $L(a),L(b)$ and the volume of $T$?

An obvious one is $L(ab) leq L(a) + L(b)$, but i guess, there are better results.

This is inspired by Loewner’s torus inequality.