Congruence hulls in SLn

The congruence subgroup problem for SLn(Z)

Following Bass–Milnor–Serre, we prove that SLn(Z) has the congruence subgroup property for n ≥ 3. This was originally proved by Mennicke and Bass–Lazard–Serre. Let Γn = SLn(Z). The congruence subgroup problem for Γn (solved independently by Mennicke [Me] and Bass–Lazard–Serre [BLS]) seeks to classify all finite-index subgroups of Γn. For l ≥ 2, the level l principal congruence subgroup of Γn, d...

CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SLn(F [t])

for all n ≥ 2. Here, we study the extent to which this isomorphism holds when the trivial Z coefficients are replaced by some rational representation of SLn(F ). The group SLn(F [t]) acts on such a representation via the map SLn(F [t]) t=0 −→ SLn(F ). There are two approaches one might take. The first is to use the spectral sequence associated to the action of SLn(F [t]) on a certain Bruhat–Tit...

CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SLn(F [t℄) II: FINITE FIELDS AND NUMBER FIELDS

Navigating in the Cayley graphs of SLN(Z) and SLN(Fp)

We give a non-deterministic algorithm that expresses elements of SLN (Z), for N ≥ 3, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the non-deterministic time-complexity of the subtractive version of Euclid’s algorithm for finding the greatest common divisor of N ≥ 3 integers a1, . . . , aN is at most a constant time...

Monomial Irreducible sln-Modules

In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.