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Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group of $T^n$ in the sequel).

Let $d$ be the distance induced by $\tilde g$ and $t:\mathbb{Z}^n\to (0,+\infty)$ be the function defined by $t(\gamma)=d(0,\gamma(0))$. Recall that the stable norm $\Vert\cdot\Vert_S$ is a norm on $\mathbb{R}^n$ defined by the property that for any $\gamma\in\mathbb{Z}^n$, $$ \Vert\gamma\Vert_S = \lim_k \frac{t(\gamma^k)}{k} $$

Question: is it true that for all $\gamma\in\mathbb{Z}^n$, we have $$t(\gamma)\le \Vert \gamma \Vert_S+2\mathrm{diam}(g)?$$

If not, does some similar control hold? The formula could depend on $n$ but not on the metric $g$.

A pointer to good literature on this kind of metric Riemannian geometry would already be much appreciated.

Important edit: if needed, I am ok with the additional, very strong assumption that the sectional curvature of $g$ is bounded above by some positive $\varepsilon=\varepsilon(n)$. The formula for a lower bound on $\Vert\cdot\Vert_S$ can depend upon $\varepsilon$ (explicitely).

As for motivation, I need this kind of control for a project of showing some constraints on Riemannian metrics on the torus $T^n$ by using quantitative versions of Milnor's argument in his paper on the growth of fundamental groups and volume of Riemannian manifolds.

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group of $T^n$ in the sequel).

Let $d$ be the distance induced by $\tilde g$ and $t:\mathbb{Z}^n\to (0,+\infty)$ be the function defined by $t(\gamma)=d(0,\gamma(0))$. Recall that the stable norm $\Vert\cdot\Vert_S$ is a norm on $\mathbb{R}^n$ defined by the property that for any $\gamma\in\mathbb{Z}^n$, $$ \Vert\gamma\Vert_S = \lim_k \frac{t(\gamma^k)}{k} $$

Question: is it true that for all $\gamma\in\mathbb{Z}^n$, we have $$t(\gamma)\le \Vert \gamma \Vert_S+2\mathrm{diam}(g)?$$

If not, does some similar control hold? The formula could depend on $n$ but not on the metric $g$.

A pointer to good literature on this kind of metric Riemannian geometry would already be much appreciated.

Important edit: if needed, I am ok with the additional, very strong assumption that the sectional curvature of $g$ is bounded above by some positive $\varepsilon=\varepsilon(n)$. The formula for a lower bound on $\Vert\cdot\Vert_S$ can depend upon $\varepsilon$ (explicitely).

As for motivation, I need this kind of control for a project of showing some constraints on Riemannian metrics on the torus $T^n$ by using quantitative versions of Milnor's argument in his paper on the growth of fundamental groups and volume of Riemannian manifolds.

Happy ending edit: while I was not able to complete the argument proposed in the accepted answer, with Stéphane Sabourau we managed to get a good enough bound, by passing to a finite cover not increasing diameter too much, see https://arxiv.org/abs/1707.07876

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group of $T^n$ in the sequel).

Let $d$ be the distance induced by $\tilde g$ and $t:\mathbb{Z}^n\to (0,+\infty)$ be the function defined by $t(\gamma)=d(0,\gamma(0))$. Recall that the stable norm $\Vert\cdot\Vert_S$ is a norm on $\mathbb{R}^n)$ defined by the property that for any $\gamma\in\mathbb{Z}^n$, $$ \Vert\gamma\Vert_S = \lim_k \frac{t(\gamma^k)}{k} $$

Question: is it true that for all $\gamma\in\mathbb{Z}^n$, we have $$t(\gamma)\le \Vert \gamma \Vert_S+2\mathrm{diam}(g)?$$

If not, does some similar control hold? The formula could depend on $n$ but not on the metric $g$.

A pointer to good literature on this kind of metric Riemannian geometry would already be much appreciated.

Important edit: if needed, I am ok with the additional, very strong assumption that the sectional curvature of $g$ is bounded above by some positive $\varepsilon=\varepsilon(n)$. The formula for a lower bound on $\Vert\cdot\Vert_S$ can depend upon $\varepsilon$ (explicitely).

As for motivation, I need this kind of control for a project of showing some constraints on Riemannian metrics on the torus $T^n$ by using quantitative versions of Milnor's argument in his paper on the growth of fundamental groups and volume of Riemannian manifolds.

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group of $T^n$ in the sequel).

Let $d$ be the distance induced by $\tilde g$ and $t:\mathbb{Z}^n\to (0,+\infty)$ be the function defined by $t(\gamma)=d(0,\gamma(0))$. Recall that the stable norm $\Vert\cdot\Vert_S$ is a norm on $\mathbb{R}^n$ defined by the property that for any $\gamma\in\mathbb{Z}^n$, $$ \Vert\gamma\Vert_S = \lim_k \frac{t(\gamma^k)}{k} $$

Question: is it true that for all $\gamma\in\mathbb{Z}^n$, we have $$t(\gamma)\le \Vert \gamma \Vert_S+2\mathrm{diam}(g)?$$

If not, does some similar control hold? The formula could depend on $n$ but not on the metric $g$.

A pointer to good literature on this kind of metric Riemannian geometry would already be much appreciated.

Important edit: if needed, I am ok with the additional, very strong assumption that the sectional curvature of $g$ is bounded above by some positive $\varepsilon=\varepsilon(n)$. The formula for a lower bound on $\Vert\cdot\Vert_S$ can depend upon $\varepsilon$ (explicitely).

As for motivation, I need this kind of control for a project of showing some constraints on Riemannian metrics on the torus $T^n$ by using quantitative versions of Milnor's argument in his paper on the growth of fundamental groups and volume of Riemannian manifolds.

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group of $T^n$ in the sequel).

Let $d$ be the distance induced by $\tilde g$ and $t:\mathbb{Z}^n\to (0,+\infty)$ be the function defined by $t(\gamma)=d(0,\gamma(1))$. Recall that the stable norm $\Vert\cdot\Vert_S$ is a norm on $C([0,\;1],\mathbb{R}^n)$ defined by the property that for any $\gamma\in\mathbb{Z}^n$, $$ \Vert\gamma\Vert_S = \lim_k \frac{t(\gamma^k)}{k} $$

Question: is it true that for all $\gamma\in\mathbb{Z}^n$, we have $$t(\gamma)\le \Vert \gamma \Vert_S+2\mathrm{diam}(g)?$$

If not, does some similar control hold? The formula could depend on $n$ but not on the metric $g$.

A pointer to good literature on this kind of metric Riemannian geometry would already be much appreciated.

Important edit: if needed, I am ok with the additional, very strong assumption that the sectional curvature of $g$ is bounded above by some positive $\varepsilon=\varepsilon(n)$. The formula for a lower bound on $\Vert\cdot\Vert_S$ can depend upon $\varepsilon$ (explicitely).

As for motivation, I need this kind of control for a project of showing some constraints on Riemannian metrics on the torus $T^n$ by using quantitative versions of Milnor's argument in his paper on the growth of fundamental groups and volume of Riemannian manifolds.

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a $\mathbb{Z}^n$-periodic metric on $\mathbb{R}^n$ (I shall conflate the lattice $\mathbb{Z}^n$ with the fundamental group of $T^n$ in the sequel).

Let $d$ be the distance induced by $\tilde g$ and $t:\mathbb{Z}^n\to (0,+\infty)$ be the function defined by $t(\gamma)=d(0,\gamma(0))$. Recall that the stable norm $\Vert\cdot\Vert_S$ is a norm on $\mathbb{R}^n)$ defined by the property that for any $\gamma\in\mathbb{Z}^n$, $$ \Vert\gamma\Vert_S = \lim_k \frac{t(\gamma^k)}{k} $$

Question: is it true that for all $\gamma\in\mathbb{Z}^n$, we have $$t(\gamma)\le \Vert \gamma \Vert_S+2\mathrm{diam}(g)?$$

If not, does some similar control hold? The formula could depend on $n$ but not on the metric $g$.

A pointer to good literature on this kind of metric Riemannian geometry would already be much appreciated.

Important edit: if needed, I am ok with the additional, very strong assumption that the sectional curvature of $g$ is bounded above by some positive $\varepsilon=\varepsilon(n)$. The formula for a lower bound on $\Vert\cdot\Vert_S$ can depend upon $\varepsilon$ (explicitely).

As for motivation, I need this kind of control for a project of showing some constraints on Riemannian metrics on the torus $T^n$ by using quantitative versions of Milnor's argument in his paper on the growth of fundamental groups and volume of Riemannian manifolds.