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 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.