There are classes of group presentations for which the conjugacy problem is known to be soluble, for example braid groups.
Is the word “soluble” here a synonym for “decidable”?
Of the groups in the soluble class, is there a particular group that is particularly simple to see why it is soluble?
Do the groups in the soluble class share a feature that causes them to be soluble?
Of Dehn’s three most important decision problems I can understand why the word problem is elementary and important (testing within a group for equality of two elements) and likewise the isomorphism problem (testing between groups for equality of themselves).
Because the word problem is a special case of the conjugacy problem it is more general and more difficult. Is this the logic behind why the conjugacy problem is also one of the three fundamental decision problems? Are there other candidates of decision problem that could be considered fundamental in group theory, or is this not a good question in some way?