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Selected Topics in Geometric Group Theory (Winter Semester 2012/13)

Change of the dates: The second lecture will be on Wednesday, 8am instead of Friday.
Problem sessions: Beginning with next week, there will be two problem sessions.

Lecture: Tuesday 11:30-13:00 Z 1
Wednesday 8:00-9:30 1C-03
Problem class: Wednesday 14:00-15:30 Z 1
Thursday 8:00-9:30 K 2

One of the main ideas of geometric group theory is to study finitely generated groups by their action on geometric spaces via symmetries and to link algebraic properties of the groups
to geometric properties of the spaces they are acting on. In this course we will study in particular the famous mapping class group, one of the main actors in this field. We will start from the scratch by the definition of the mapping class group. See the announcement for more information about the content.

Problem sheets
Sheet 1
Sheet 2
Sheet 3 There was a problem with one of the questions. We will resolve this in the next problem session.
Sheet 4
Sheet 5 Ideas for solving these problems will be presented only after two weeks. There is nevertheless a problem session!
Sheet 6
Sheet 7
Sheet 8
Sheet 9
Sheet 10
Sheet 11

Prerequisites: Basic knowledge in geometry, topology and algebra e.g. from the courses Grundlagen in Geometrie und Topologie and Grundlagen in Algebra und Zahlentheorie.

Notes for the second step in the proof of the bigon criterion


Books on the topic of the course

J. Birman: Braids, Links And Mapping Class Groups, Princeton University Press 1974.
B. Farb and D. Margalit: A Primer on Mapping Class Groups, Princeton University Press.
B. Farb: Problems on Mapping Class Groups and Related Topics, Proc. Symp. Pure and Applied Math., Volume 74, 2006.
N. Ivanov: Subgroups of Teichmüller Modular Groups, AMS 1992.

Additional books on the background

L. Ahlfors und L. Sario: Riemann Surfaces, Princeton University Press 1960.
W. Fulton: Algebraic Topology - A First Course, Springer 1995.
J. Hubbard: Teichmüller Theory, Ithaca, NY: Matrix Editions (http://MatrixEditions.com), 2006.
Y.Imayoshi, M.Taniguchi: "An introduction to Teichmüller Spaces", Springer 1992.
J. Lee: Introduction to Smooth Manifolds, Springer 2013.
W.S. Massey: Algebraic topology: An introduction, Springer 1967.

Additional articles

R. Baer: Kurventypen auf Flächen, J. Reine Angew. Math. 156 (231-246), 1927.
R. Baer: Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen, J. Reine Angew. Math. 159 (101-111), 1928.
D. Dumas: Complex Projective Structures, Handbook of Teichmüller theory. Volume II (EMS). IRMA Lectures in Mathematics and Theoretical Physics 13, 455-508 (2009).
D. Gale: The classification of 1-manifolds: a take-home exam., The American Mathematical Monthly, 94(2):170, 1987.
H. R. Morton: The space of homeomorphisms of a disc with n holes, Illinois J. Math. Volume 11, Issue 1 (1967), 40-48.

Introductory lecture notes of other courses (all in German)

Stefan Kühnlein: Einführung in die Geometrie und Topologie
Studentischer Vorlesungsmitschrieb - Version 1 and Version 2 of the lecture Einführung in die Geometrie und Topologie of Wilderich Tuschmann