Math 9071: Teichmuller and Nielsen-Thurston Theory

Spring 2017

 Meets: Tue/Thu 2:00 - 3:200 in Wachman Hall, room 1036 Instructor: David Futer Office: 1026 Wachman Hall Office Hours: by appointment E-mail: dfuter at temple.edu Phone: (215) 204-7854

Course content: This course will focus on the symmetry groups of manifolds and cell complexes. We will look at both the geometric and algebraic properties of the mapping class group, which can roughly be thought of as the group of symmetries of a surface. One of the ways in which we will study the group is by designing a certain cell complex on which it acts by isometries. This principle has wide generalizations: one can derive a great deal of information about a group by constructing a geometric object for the group to act on.

In the spring semester, we will focus our attention on the braid groups, which are a special class of mapping class groups. In particular, we will study the representation theory of these groups, with connections to invariants of knots and links.

Textbooks:

Prerequisites: Math 9120.

Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.

Detailed schedule

1/17 Residual finiteness P. 177-181 See Alperin paper
1/19 Torsion P. 200-203
1/24 Orbifold basics P.203-206
1/26 Geometric orbifolds; Hurwitz's theorem P. 207-210
1/31 Wyman's theorem; realizing finite groups P.211-214
2/2 Dehn-Nielsen-Baer theorem P.219-221, 236-238
2/7 Braid groups P. 239-247
2/9 no class
2/14 Intro to Teichmuller space P. 261-267
2/16 Teichmuller space via lattices P. 269-272
2/21 Dimension counts; hexagons and pants P. 272-277
2/23 Fenchel-Nielsen coordinates P. 278-285 Homework 1, due 3/2
3/2 Teichmuller geometry P. 294-298
3/7 Quasiconformal maps, measured foliations P. 299-304
3/9 Singular measured foliations P. 305-308
3/21 Holomorphic quadratic differentials P. 309-313
3/23 The vector space of quadratic differentials P. 314-318
3/28 Teichmuller maps P. 319-324
3/30 Grotzcsh's theorem; Teichmuller uniqueness P. 325-330
4/4 Teichmuller existence, geodesics P. 331-339
4/6 Moduli space; Fricke's theorem P. 342-351
4/11 Mumford's compactness criterion P. 351-357
4/13 Three types of mapping classes P. 367-369, 374-376
4/18 Periodic mapping classes; reducing systems P. 370-373
4/20 Collar lemma; parabolic implies reducible P. 380-385 Homework 2, due 5/5
4/25 Hyperbolic isometries are pseudo-Anosov P. 385-388
4/27 Casson's criterion; lengths under iteration P. 397-400, 419-422

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dfuter at temple edu