The path $\ell$ is not well defined. That is, there is no best path of pleated surfaces around a mapping torus. Thus the answer to your question is "no": there is no upper bound on the Hausdorff distance between $\ell$ and $\sigma$, purely in terms of the topology of $S$.

To prove this, we resort to a standard example. Suppose that $S$ is the genus two surface. Suppose $X$ and $Y$ are disjoint subsurfaces of $S$, both homeomorphic to the once-holed torus. Let $\gamma = \partial X = \partial Y$.

Let $f$ be a pA map on $X$, let $g$ be a pA map on $Y$, and let $h$ be a pA map on $S$. Then for all sufficiently large $n$ the map $F_n = (f g)^n h$ is a pA map. Let $M_n$ be the mapping torus for $F_n$. Then the curve $\delta = \gamma \times \{1/2\} \subset M_n$ is very short in the hyperbolic metric in $M_n$. Consider the following two paths $\ell$ and $\ell'$ of pleated surfaces. The path $\ell$ first "moves through" $X$ and then moves through $Y$. The path $\ell'$ moves through $X$ and $Y$ in the opposite order. The paths $\ell$ and $\ell'$ are far apart in Teichmüller space, so at least one of them is far away from $\sigma$, as measured in Hausdorff distance. (The exact same construction applies to the WP metric as well.)

Basically, the thin part of Teichmüller space contains large product-like regions. Teichmüller geodesics "know" how to go through such regions. Pleating paths, which are very far from being unique, do not have such knowledge. As a final remark - there are periodic Teichmüller geodesics that live completely in the thin part. Using such a $\sigma$ we can arrange an $\ell$ that is at no point close to $\sigma$.

The path $\ell$ is not well defined. That is, there is no best path of pleated surfaces around a mapping torus. Thus the answer to your question is "no": there is no upper bound on the Hausdorff distance between $\ell$ and $\sigma$, purely in terms of the topology of $S$.

To prove this, we resort to a standard example. Suppose that $S$ is the genus two surface. Suppose $X$ and $Y$ are disjoint subsurfaces of $S$, both homeomorphic to the once-holed torus. Let $\gamma = \partial X = \partial Y$.

Let $f$ be a pA map on $X$, let $g$ be a pA map on $Y$, and let $h$ be a pA map on $S$. Then for all sufficiently large $n$ the map $F_n = (f g)^n h$ is a pA map. Let $M_n$ be the mapping torus for $F_n$. Then the curve $\delta = \gamma \times \{1/2\} \subset M_n$ is very short in the hyperbolic metric in $M_n$. Consider the following two paths $\ell$ and $\ell'$ of pleated surfaces. The path $\ell$ first "moves through" $X$ and then moves through $Y$. The path $\ell'$ moves through $X$ and $Y$ in the opposite order. The paths $\ell$ and $\ell'$ are far apart in Teichmüller space, so at least one of them is far away from $\sigma$, as measured in Hausdorff distance. (The exact same construction works if we instead use the WP metric.)

Basically, the thin part of Teichmüller space contains large product-like regions. Teichmüller geodesics "know" how to go through such regions. Pleating paths, which are very far from being unique, do not have such knowledge. As a final remark - there are periodic Teichmüller geodesics that live completely in the thin part. Using such a $\sigma$ we can arrange an $\ell$ that is at no point close to $\sigma$.

The path $\ell$ is not well defined. That is, there is no best path of pleated surfaces around a mapping torus. Thus the answer to your question is "no": there is no upper bound on the Hausdorff distance between $\ell$ and $\sigma$, purely in terms of the topology of $S$.

To prove this, we resort to a standard example. Suppose that $S$ is the genus two surface. Suppose $X$ and $Y$ are disjoint subsurfaces of $S$, both homeomorphic to the once-holed torus. Let $\gamma = \partial X = \partial Y$.

Let $f$ be a pA map on $X$, let $g$ be a pA map on $Y$, and let $h$ be a pA map on $S$. Then for all sufficiently large $n$ the map $F_n = (f g)^n h$ is a pA map. Let $M_n$ be the mapping torus for $F_n$. Then the curve $\delta = \gamma \times \{1/2\} \subset M_n$ is very short in the hyperbolic metric in $M_n$. Consider the following two paths $\ell$ and $\ell'$ of pleated surfaces. The path $\ell$ first "moves through" $X$ and then moves through $Y$. The path $\ell'$ moves through $X$ and $Y$ in the opposite order. The paths $\ell$ and $\ell'$ are far apart in Teichmüller space, so at least one of them is far away from $\sigma$, as measured in Hausdorff distance. (The exact same construction applies to the WP metric as well.)

Basically, the thin part of Teichmüller space contains large product-like regions. Teichmüller geodesics "know" how to go through such regions. Pleating paths, which are very far from being unique, do not have such knowledge.

The path $\ell$ is not well defined. That is, there is no best path of pleated surfaces around a mapping torus. Thus the answer to your question is "no": there is no upper bound on the Hausdorff distance between $\ell$ and $\sigma$, purely in terms of the topology of $S$.

To prove this, we resort to a standard example. Suppose that $S$ is the genus two surface. Suppose $X$ and $Y$ are disjoint subsurfaces of $S$, both homeomorphic to the once-holed torus. Let $\gamma = \partial X = \partial Y$.

Let $f$ be a pA map on $X$, let $g$ be a pA map on $Y$, and let $h$ be a pA map on $S$. Then for all sufficiently large $n$ the map $F_n = (f g)^n h$ is a pA map. Let $M_n$ be the mapping torus for $F_n$. Then the curve $\delta = \gamma \times \{1/2\} \subset M_n$ is very short in the hyperbolic metric in $M_n$. Consider the following two paths $\ell$ and $\ell'$ of pleated surfaces. The path $\ell$ first "moves through" $X$ and then moves through $Y$. The path $\ell'$ moves through $X$ and $Y$ in the opposite order. The paths $\ell$ and $\ell'$ are far apart in Teichmüller space, so at least one of them is far away from $\sigma$, as measured in Hausdorff distance. (The exact same construction applies to the WP metric as well.)

Basically, the thin part of Teichmüller space contains large product-like regions. Teichmüller geodesics "know" how to go through such regions. Pleating paths, which are very far from being unique, do not have such knowledge. As a final remark - there are periodic Teichmüller geodesics that live completely in the thin part. Using such a $\sigma$ we can arrange an $\ell$ that is at no point close to $\sigma$.