Geometry of Nilpotent and Solvable Groups

Taught Course Centre (January-March 2011)

Time: Wednesday 11 am -1 pm. The TCC timetable can be found here.

Syllabus: I plan to cover the following topics, listed in the order of their appearance in lectures.

Course content: The course focuses on nilpotent and solvable groups. It takes a modern geometric approach to these groups rather than a classical combinatorial one.

A standard way of endowing a group with a geometry is through the Cayley graph and its associated metric (the word metric). A main question in this setting is to recover as much as possible algebraic information from the geometry of the Cayley graph (in other words "can you see the algebra of a group ?"). Several striking results state that strong algebraic properties may be recognised via loose geometric features. One of these is the equivalence established by the Bass Theorem, the Milnor-Wolf Theorem and the Gromov Polynomial Growth Theorem: a group is nilpotent (up to finite index) if and only if the number of vertices in the Cayley graph contained in a ball grows as a polynomial function of the radius.

One may also try to understand a group by representing it as a group of matrices, i.e. as a linear group. Jacques Tits' Alternative Theorem came as a positive answer to the celebrated von Neumann-Day question, in the setting of linear groups. But it can also be viewed as a striking characterization of solvable linear groups in the class of all linear groups.

Lecture notes:

Lecture 1. These notes are posted in a LaTeX-ed version.

Lecture 2. These notes are posted in a LaTeX-ed version.

Lecture 3.

Lecture 4.

Lecture 5.

Lecture 6.

Lecture 7.

Recommended reference for algebraic groups (unfortunately out of print): the book of Onishchik and Vinberg.

Lecture 8. These notes are posted in a LaTeX-ed version.

Lecture 9. These notes are posted in a LaTeX-ed version.

Course Assessment: The course is assessed via problem sheets posted here.

Exercise Sheet 1.

Exercise Sheet 2. The solutions to this Exercise Sheet are to be sent to Cornelia Drutu by Monday February 28. They can be sent either by e-mail or by ordinary mail.

References: The material of this course (lectures and exercises) is taken, for the most part, from the forthcoming book of C. Drutu and M. Kapovich, "Lectures on Geometric Group Theory".

The course relies on the seminal works of M. Gromov (Abel prize 2009) and of J. Tits (Abel prize 2008). The book "Asymptotic invariants of infinite groups" of M. Gromov is the main reference book. The proofs that are not provided in "Asymptotic invariants" are taken from various papers that appeared later on.

The papers containing the initial proofs of several main theorems in the syllabus are listed below. Still, in some cases the course will present more recent versions of these proofs. For instance, the proof of Jacques Tits' Alternative Theorem explained in lectures will also use the work of E. Breuillard and T. Gelander below, etc.

  • M. Gromov, "Asymptotic invariants of infinite groups", in "Geometric Group Theory", Vol. 2 (Sussex, 1991), G. Niblo and M. Roller (eds), LMS Lecture Notes, volume 182, Cambridge Univ. Press, 1993.
  • M. Gromov, "Groups of polynomial growth and expanding maps", Publ. Math. IHES 53 (1981), 53-73.
  • H. Bass, "The degree of polynomial growth of finitely generated nilpotent groups", Proc. LMS 25 (1972), 603-614.
  • E. Breuillard and T. Gelander, "On dense free subgroups of Lie groups", J. Algebra 261 (2003), 448-467.
  • J. Milnor, "A note on curvature and fundamental group", J. Diff. Geom. 2 (1968), 1-7.
  • J. Tits, "Free subgroups in linear groups", Journal of Algebra 20 (1972), 250-270.
  • J.A. Wolf, "Growth of finitely generated solvable groups and curvature of Riemannian manifolds", Journal Diff. Geom. 2 (1968), 421-446.