The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk.

What kinds of quantitative results are known about this problem? Specifically, what are the known best upper bound estimates for the time and space complexity of these algorithms?

Suppose we use a triangulation as genie. Are there any results like the following: given a 3-manifold $M$ and a triangulation $T$ then there is a PL loop $\gamma$ of length $l$ in $M$ such that any triangulation $T'$ compatible with a disk bounding this loop must be of distance greater than $e^l$ from $T$.

The word problem for 3-manifold groups is solvable: given a based loop $\gamma$ in $M^3$ there is an algorithm to decide whether $\gamma$ bounds a disk.

What kinds of quantitative results are known about this problem? Specifically, what are the known best upper bound estimates for the time and space complexity of these algorithms?

Suppose we use a triangulation as a genie, are there any results like the following? If $(M,T)$ is a closed 3-manifold $M$ and a triangulation $T$ of $M$ then there exists a PL loop $\gamma$ in $M$ such that if $T'$ is a triangulation of $M$ with a 2-skeleton containing a disk bounding the loop then the Pachner distance between the two triangulations $T$ and $T'$ must be greater than $e^l$, where $l$ is the length of $\gamma$.