Bounded generation of some linear groups

Morgan, Aleksander, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Rapinchuk, Andrei, AS-Mathematics, University of Virginia

Let R be a commutative ring with identity. For n >= 2, we let E_n(R) denote the
subgroup of the special linear group SL_n(R) generated by all elementary matrices, and for an ideal
J of R we let E_n(R; J) denote the normal subgroup of E_n(R) generated by the elementary matrices that are congruent to the identity matrix modulo J. The goal of this dissertation is to establish several
results asserting that in certain situations the groups E_n(R) and E_n(R; J) have finite width with
respect to their natural generating sets. In particular, it is shown in Chapter II that if R is a ring
of algebraic S-integers with infinitely many units, then E_2(R) (which in this case coincides with
SL_2(R)) has width <= 9 with repect to elementary matrices. This result yields bounded generation
of SL_2(R) in the case at hand, as an abstract group. The results of Chapter III apply to the ideals
J in an arbitrary Noetherian ring R for which the quotient R=J is finite. We show that if E_n(R) has
finite width with respect to elementaries, then every element in E_n(R; J^2) is a product of a bounded
number of elementary matrices congruent to In modulo J. Assuming further that R satisfies Bass's
stable range condition, we derive from this result that the normal subgroup of E_n(R) generated
by a given matrix subject to some conditions, has finite index and finite width with respect to the
union of the conjugacy classes of the matrix and its inverse.

PHD (Doctor of Philosophy)
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