Elementary matrix operations on a napkin

Department of Mathematical Sciences Colloquium presented by Dr. Bogdan Nica, Indiana University-Purdue University Indianapolis.

Matrices of determinant 1 can be reduced to the identity matrix by row- and column-operations. This is a well-known fact of linear algebra–assuming that we work with real or complex entries. A closer look reveals that the number of elementary operations needed to reduce an n-by-n matrix can be estimated as a function of n, independently of the matrix! (Can you find such a function?) This is a phenomenon of `bounded elementary reduction’ or, to put it differently, of `bounded elementary generation’.

It is easy to ask, but hard to answer, whether an analogous `bounded elementary generation’ holds for matrices with integral entries (meaning that the allowed elementary operations have to be integral as well). This is now a group-theoretic question with a strong number-theoretic flavor; we have definitely left Kansas. The main goal of my talk is to give a friendly overview of `bounded elementary generation’ over the integers, and survey its status for other rings of arithmetic nature.

Bounded elementary generation has been linked with algebraic K-theory, Kazhdan’s property (T), representation rigidity, the congruence subgroup property, and bounded cohomology. Time permitting, I will touch on the latter.