The fundamental group of an affine manifold. Mathematical developments arising from Hilbert 18th problem.

Speaker
Gregory Soifer, Bar-Ilan University and Weizmann Institute
Date
04/06/2017 - 15:00 - 14:00Add to Calendar 2017-06-04 14:00:00 2017-06-04 15:00:00 The fundamental group of an affine manifold. Mathematical developments arising from Hilbert 18th problem. The study of affine crystallographic groups has a long history which goes back to  Hilbert's 18th problem. More precisely  Hilbert  (essentially) asked  if there is only a finite number, up to conjugacy in  Aff(R^n) of crystallographic groups G acting  isometrically on R^n. In  a series  of  papers  Bieberbach  showed that this was so. The key result is the following  famous theorem of Bieberbach. A crystallographic group G acting isometrically on  the n-dimensional Euclidean space R^n contains a subgroup  of finite index consisting of translations. In particular, such a group G is virtually abelian, i.e. G contains an abelian subgroup of finite index. In 1964 Auslander proposed  the following conjecture  The Auslander Conjecture: Every crystallographic subgroup G of Aff(R^n) is virtually solvable, i.e. contains a solvable subgroup of finite index.  In 1977 J. Milnor stated the following question: Question: Does there exist a complete affinely flat manifold M such that the fundamental group of M contains a free group? We will explain ideas and methods, recent and old  results related to the above problems.    Colloquium Room 201, Bldg 216 אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Colloquium Room 201, Bldg 216
Abstract

The study of affine crystallographic groups has a long history which goes back to  Hilbert's 18th problem. More precisely  Hilbert  (essentially) asked  if there is only a finite number, up to conjugacy in  Aff(R^n) of crystallographic groups G acting  isometrically on R^n. In  a series  of  papers  Bieberbach  showed that this was so. The key result is the following  famous theorem of Bieberbach. A crystallographic group G acting isometrically on  the n-dimensional Euclidean space R^n contains a subgroup  of finite index consisting of translations. In particular, such a group G is virtually abelian, i.e. G contains an abelian subgroup of finite index. In 1964 Auslander proposed  the following conjecture 

The Auslander Conjecture: Every crystallographic subgroup G of Aff(R^n) is virtually solvable, i.e. contains a solvable subgroup of finite index. 

In 1977 J. Milnor stated the following question:

Question: Does there exist a complete affinely flat manifold M such that the fundamental group of M contains a free group?

We will explain ideas and methods, recent and old  results related to the above problems. 
 

Last Updated Date : 28/05/2017