Series: Mathematics Colloquium

Date: Thursday, May 1, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Joseph Hundley

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: Dorian Goldfeld, Columbia University

Title: Counting Congruence Subgroups

Abstract: 

Subgroup growth studies the distribution of subgroups of finite index
in a group as a function of the index.  In the last two decades this
topic has developed into one of the most active areas of research in
infinite group theory.  There is a growing realization that there are
deep connections between subgroup growth of a group and the algebraic
structure of that group.  For example the so-called PSG Theorem
characterizes the groups of polynomial subgroup growth as those which
are virtually soluble of finite rank.  A key element in the proof is
the growth of congruence subgroups in arithmetic groups, a new kind of
"non-commutative arithmetic", with applications to the study of
lattices in Lie groups.  Roughly, arithmetic groups are matrix groups
over the ring of integers of an algebraic number field.  An arithmetic
group is a congruence group if the entries of each matrix in the group
satisfy some elementary congruence conditions such as a particular
entry is divisible by a fixed integer.  In joint work with Lubotzky
and Pyber, we obtain asymptotic formulae (as n goes to infinity) for
the number of congruence subgroups of index at most n in SL(2,Z).  I
will also indicate some generalizations of these results to SL(r,O_k)
with r > 2, where O_k denotes the ring of integers in an algebraic
number field, k.