Congruence subgroup problems

Geometry Topology Seminar
Monday, September 10, 2012 - 14:05
1 hour (actually 50 minutes)
Skiles 006
U Wisconsin
It is a theorem of Bass, Lazard, and Serre, and, independently, Mennicke, that the special linear group SL(n,Z) enjoys the congruence subgroup property when n is at least 3.  This property is most quickly described by saying that the profinite completion of the special linear group injects into the special linear group of the profinite completion of Z.  There is a natural analog of this property for mapping class groups of surfaces.  Namely, one may ask if the profinite completion of the mapping class group embeds in the outer automorphism group of the profinite completion of the surface group. M. Boggi has a program to establish this property for mapping class groups, which couches things in geometric terms, reducing the conjecture to determining the homotopy type of a certain space.  I'll discuss what's known, and what's needed to continue his attack.