Background
The notion of an absolute presentation arises from Bernhard Neumann's study of the isomorphism problem for algebraically closed groups.
A common strategy for considering whether two groups and are isomorphic is to consider whether a presentation for one might be transformed into a presentation for the other. However algebraically closed groups are neither finitely generated nor recursively presented and so it is impossible to compare their presentations. Neumann considered the following alternative strategy:
Suppose we know that a group with finite presentation can be embedded in the algebraically closed group then given another algebraically closed group, we can ask "Can be embedded in ?"
It soon becomes apparent that a presentation for a group does not contain enough information to make this decision for while there may be a homomorphism, this homomorphism need not be an embedding. What is needed is a specification for that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.
Read more about this topic: Absolute Presentation Of A Group
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