# C2.4 Infinite Groups - Material for the year 2019-2020

## Primary tabs

Knowledge of the first and second-year algebra courses is helpful but not mandatory; in particular Prelims Groups and Group Actions, A0 Linear Algebra, and ASO Group Theory. Likewise, the course B3.5 Topology and Groups would bring more familiarity and a different viewpoint of the notions treated in this course.

16 lectures.

### Assessment type:

- Written Examination

The course introduces some natural families of groups, with an emphasis on those that generalize abelian groups, various questions that one can ask about them, and various methods used to answer these questions; these involve among other things questions of finite presentability, linearity, torsion and growth.

Free groups; ping-pong lemma. Finitely generated and finitely presented groups. Residual finiteness and linearity.

Nilpotency, lower and upper central series. Polycyclic groups, length, Hirsch length, Noetherian induction. Solvable groups, derived series, law.

Structure of linear solvable and nilpotent groups.

Solvable versus polycyclic: examples, characterization of polycyclic groups as solvable Z-linear groups.

Solvable versus nilpotent: the Milnor -Wolf theorem characterizing nilpotent groups as solvable groups with sub-exponential (and in fact polynomial) growth.

- C. Drutu, M. Kapovich,
*Geometric Group Theory*, (AMS, 2018), Chapters 13 and 14. - D. Segal,
*Polycyclic groups*, (CUP, 2005) Chapters 1 and 2. - D. J. S. Robinson,
*A course in the theory of groups*, 2nd ed., Graduate texts in Mathematics, (Springer-Verlag, 1995). Chapters 2, 5, 6, 15.