Advanced Topics in Geometry - Geometric Group Theory
Change of plans: the exam will be written. More info here.
Lectures: Tuesdays 4-6pm and Thursdays 2-4pm, starting October 27. Course pages on BASIS and eCampus.
Links to Zoom room and password available on eCampus. Lecture pdfs available on eCampus and below. Version with bookmarks created by Luis Henriques available here.
Course description: Geometric group theory is the study of the algebraic properties of finitely generated, infinite groups via their isometric actions on metric spaces.
Basic topics that I will discuss: Cayley graphs, quasi-isometries, growth in groups, groups acting on trees, Gromov-hyperbolic groups and their boundaries at infinity. Depending on time, possible more advanced topics for the end of the course include Mostow rigidity, Gromov's polynomial growth theorem, the Tits alternative.
Prerequisites: elementary knowledge of group theory, topology (especially: covering spaces) and metric spaces.
References:
C. Löh, Geometric Group Theory: An Introduction, Springer, 2017. Draft and errata freely available (if your eyes are feeling brave).
Office Hours with a Geometric Group Theorist, edited by M. Clay and D. Margalit, Princeton University Press, 2017.
C. Druţu and M. Kapovich, Geometric Group Theory, American Mathematical Society, 2018. Draft available here.
M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, 1999. A pdf of the book available here.
B. Bowditch, A course on geometric group theory, 2005.
A. Sisto, Lecture notes on geometric group theory, 2014.
J.-P. Serre, Trees, Springer, 2003.
Past lectures:
-- 27.10 -- §1 Generating sets, §2 Free groups. pdf (updated 30.10)
-- 29.10 -- §2 Free groups, §3 Group presentations. pdf
-- 03.11 -- §3 Group presentations, §4 Cayley graphs. pdf
-- 05.11 -- §4 Cayley graphs, §5 Quasi-isometries. pdf (updated 18.11)
-- 10.11 -- §5 Quasi-isometries. pdf (updated 12.11)
-- 12.11 -- §5 Quasi-isometries, §6 The Milnor-Schwarz lemma. pdf
-- 17.11 -- §6 The Milnor-Schwarz lemma, §7 Subgroup distortion. pdf
-- 19.11 -- §7 Subgroup distortion, §8 Growth. pdf
-- 24.11 -- §8 Growth. pdf
-- 26.11 -- §9 Nilpotent and solvable groups, §10 Reminder on hyperbolic geometry. pdf (updated 08.12)
-- 01.12 -- §10 Reminder on hyperbolic geometry. pdf
-- 03.12 -- §10 Reminder on hyperbolic geometry, §11 Gromov-hyperbolic spaces. pdf
-- 08.12 -- §11 Gromov-hyperbolic spaces. pdf
-- 10.12 -- §11 Gromov-hyperbolic spaces, §12 Length. pdf
-- 15.12 -- §13 Proof of the Morse lemma, §14 Properties of hyperbolic groups. pdf
-- 17.12 -- §14 Properties of hyperbolic groups. pdf
-- 22.12 -- §14 Properties of hyperbolic groups. pdf
-- 07.01 -- §14 Properties of hyperbolic groups. pdf
-- 12.01 -- §14 Properties of hyperbolic groups, §15 Rips complexes. pdf
-- 14.01 -- §15 Rips complexes, §16 The Gromov boundary. pdf
-- 19.01 -- §16 The Gromov boundary. pdf
-- 21.01 -- §16 The Gromov boundary. pdf
-- 26.01 -- §16 The Gromov boundary. pdf
-- 28.01 -- §16 The Gromov boundary, §17 Dynamics on the Gromov boundary. pdf
-- 02.02 -- §17 Dynamics on the Gromov boundary. pdf
-- 04.02 -- §18 Group amalgams and Bass-Serre theory. pdf
-- 09.02 -- §18 Group amalgams and Bass-Serre theory. pdf
-- 11.02 -- §18 Group amalgams and Bass-Serre theory. pdf