MA4H4 : Geometric group theory


Monday, 13.00, B1.01.
Tuesday, 15.00, B3.03.
Thursday, 12.00, B3.03.
No Lecture on Friday (despite what the official timetable says!)

Support class with Alex Wendland.
Tuesday 17.00, MS.03.


Group presentations. Cayley graphs. Quasi-isometries. (Fundamental groups and hyperbolic geometry.) Hyperbolic groups. Isoperimetric inequalities. [Other topics depending on time and demand.]


100% written examination.
You can find past exam papers here: here. (You need to log in!)
Note: the 2014 paper was set by a different lecturer, and based on a slightly different syllabus from other years.
In 2016, the examination will have the form of one compulsory question worth 40%, and a choice of three out of four optional questions worth 20% each.


The following supplement the material of my MSJ notes (subject to modification, see lectures).

Part 1 Supplement to Section 6 (word problem).
Part 2 First part of Section 7 (isoperimetric stuff).
Part 3 Second part of Section 7.

Exercise sheets

Prepared with the help of Robert Tang.

Sheet 1
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Sheet 6



B.H.Bowditch [me]: A course on geometric group theory. MSJ Memoirs Volume 16. Mathematical Society of Japan, 2006.
[These are the notes of a course I gave in Tokyo. They will cover a large chunk of the course.]

P. de la Harpe, Topics in geometric group theory : Chicago lectures in mathematics, University of Chicago Press (2000).
[Includes many interesting topics in GGT.]

M. Bridson, A. Haefliger, Metric spaces of non-positive curvature : Grundlehren der Math. Wiss. No. 319, Springer (1999).
[Not an introduction to GGT as such, but it has accounts of many of the main topics we will be covering.]

W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory: presentations of groups in terms of generators and relations : Interscience (1966).
[A more traditional approach to combinatorial group theory.]

R.C. Lyndon, P. Schupp, Combinatorial group theory : Springer (1977).

E.Ghys, A.Haefliger, A.Verjovsky, (eds.) Group theory from a geometrical viewpoint World Scientific.
[The "Trieste notes". A collection of articles, some expository, from the early days of GGT. Out of print, sadly.]

Books/articles on hyperbolic groups.

Several accounts were written around shorly after Gromov's paper. There's been no systematic general introduction since (for some reason).

E.Ghys, P. de la Harpe eds. Sur les groupes hyperboliques d'après Mikhael Gromov : Progress in Mathematics No. 83, Birkhauser (1990).
[The most commonly cited introduction. Mostly in French.]

M.Coornaert, T.Delzant, A.Papadopoulos, Les groupes hyperboliques de Gromov : Lecture notes in Mathematics No. 1441, Springer Verlag (1990).
[Another perspective. Also in French...]

See also the notes by Short et al, and by Bowditch (me) in the Trieste notes above.

Web links

Jon McCammond's Geometric group theory page lists lots of famous people, conferences, etc.

A truncated icosidodecahedron (The Cayley graph of the isosahedral group after collapsing double edges.)

Hyperbolic planar tessellations by Don Hatch (celebrated composer of Manfred the murderer).

A tessellation of hyperbolic 3-space by right-angled regular dodecaheda. This isn't quite the same the one you get from Seifert-Weber space. That one has five dodecahedra meeting around each edge rather than four.