# Spring 2014 MATH 257 001 LEC

Section | Days/Time | Location | Instructor | CCN |
---|---|---|---|---|

001 LEC | MWF 11-12P | 87 EVANS | AGOL, I | 54438 |

Units/Credit | Final Exam Group | Enrollment |
---|---|---|

4 | NONE | Limit:35 Enrolled:10 Waitlist:0 Avail Seats:25 [on 05/28/14] |

**Prerequisites:** 250A.

**Syllabus: Examples of Groups**

This topics course will be a tour of the zoo of infinite groups, usually endowed with some extra properties or structure. Examples include:

- Combinatorics: small-cancellation groups, Burnside groups, nilpotent groups
- Topology: fundamental groups, topological, diffeomorphism, and mapping class groups
- Geometry: CAT(0) and hyperbolic groups, Coxeter and Artin groups, K ̈ahler groups, Lie groups, algebraic groups, lattices
- Profinite: p-adic analytic and Golod-Shafarevich groups
- Number theory: Arithmetic groups, monodromy groups, Galois groups
- Dynamics: automatic and automata groups, iterated monodromy groups

Of course, many of these examples fall into many categories.

We will also discuss properties of and operations on groups, such as residual finiteness, growth, amenability, property (T), rank, presentations and finiteness conditions, word problem, cohomology, subgroups, boundaries, varieties of groups, different types of products. In addition to describing examples and their properties, we hope to highlight some of the key achievements in the classification of the various flavors of groups, such as Margulis superrigidity, classification of groups of cohomological dimension one, Gromov’s polynomial growth theorem, the Burnside problem, and the openness of finite-index subgroups of profinite groups. Open problems will be discussed as well.

Topics covered will depend on the interests of the participants, and we will likely not delve into the classification of finite groups except what is needed in the service of profinite groups.

**Office:** 921 Evans Hall

**Office Hours:** M F 10-11, W 12-1

**Required Text:** NA

**Recommended Reading:** see course webpage; Montgomery-Zippin, Topological Transformation Groups

**Grading:**

**Homework:**