[Math] Infinite group theory recommendations

abstract-algebrabook-recommendationgroup-theoryinfinite-groupsreference-request

What is a good book to start a journey in the field of infinite group theory ? I have already taken a first course in algebra where we studied the most important (finite) algebraic structures and I'm taking the second course so I'm used to the basic tools of abstract algebra, however Infinite groups (except for $\mathbb{Z}$ of course) has only been cited as examples and never studied in details so I'd like a text to start the topic but that also focuses on it (without the "finite group theory" part). I'd like a book as general as possible but if I have to choose a particular kind of groups I guess Linear Groups are a good point to start (as I already encountered them in other courses).

Best Answer

As said by @PaulPlummer, there is no general theory of infinite groups.

There are nonetheless some special theories that are still quite broad and are very important. They do tend to tie in with other branches of mathematics, though.

Beyond your idea of linear groups I'll make just a couple of suggestions:

  • Lie groups (with close ties to differential geometry).
  • Coxeter groups (these are rather special and yet are a pre-requisite to a deeper understanding of both linear groups and Lie groups).
  • Combinatorial group theory (which is closely related to topology via the fundamental group of a topological space).
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