Is a rigger the same as a dominant?

No. A rigger ties. That's a craft, and it doesn't imply any authority at all.

Plenty of rope is collaborative: two people working on something together, with the person being tied giving constant feedback about what their hands are doing. Some riggers lead; many are absorbed in the technical problem instead.

This matters practically, because a rope bunny who assumes the rigger is in charge of everything is less likely to speak up about numbness. In rope, speaking up is the whole safety system.