I am trying to understand the intuition behind recursively presented groups, and as a corollary why they are important or useful.
Here are some questions that hopefully will aid understanding, of which the first is not specific to recursively presented groups.
For a group that needs to have an infinite number of relations (that is excluding superfluous relations that can be obtained by other included relations) it is necessary for it to have an infinite number of generators?
Informally, it seems that a recursively presented group can have countably infinite generators, and the map between the natural numbers to the generators should be a recursive set, and the same must apply to the relations. Is this correct?
Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. This seems trivial since if the recursively presented group has an infinite set of generators then it cannot be finitely presented?
The group of integers under addition is a recursively presented group but not a finitely presented group?
What other good (simple / important) examples are there of recursively presented groups, with particular interest in those that will aid in understanding the concept?
Why are recursively presented groups important, is it because informally anything larger than them will more likely be intractable in a computability theory setting?