markiv said:What's an example of a group that has finitely many generators, but cannot be presented using only finitely many relations? Are there any nice groups? They do exist, right?
An Infinitely Presented, Finitely Generated Group (IPFG) is a type of mathematical object that is used to study the structure and properties of groups. It is defined as a group that has an infinite number of generators and can be presented by a finite set of defining relations.
An IPFG differs from a finitely presented group in that it has an infinite number of generators, while a finitely presented group has a finite number of generators. This means that an IPFG has a more complex structure and can have more complicated properties.
Some examples of IPFGs include the Baumslag-Solitar groups, the Higman-Neumann-Neumann groups, and the Thompson groups. These groups have been studied extensively in mathematics due to their interesting and unique properties.
IPFGs have applications in a variety of fields, including algebraic topology, number theory, and geometric group theory. They can also be used to study the fundamental group of topological spaces and to classify certain types of groups.
Some open questions in the study of IPFGs include the classification of all IPFGs, the existence of finitely presented IPFGs, and the relationship between different properties of IPFGs, such as amenability and residual finiteness.