Bounded generation for congruence subgroups of Sp4(R)

TLDR: In this paper , a bounded generation result concerning the minimum number of conjugates of suitable elementary matrices (or more precisely root elements) needed to write any element of the principal congruence subgroup of a ring of algebraic integers in any number field is given.

Abstract: For rings of algebraic integers [Formula: see text] in a number field [Formula: see text] called [Formula: see text]-pseudo-good, this paper describes a bounded generation result concerning the minimal number of conjugates of suitable elementary matrices (or more precisely root elements) in [Formula: see text] needed to write any element of the [Formula: see text]-principal congruence subgroup of [Formula: see text] as their product. Using this bounded generation result, we give explicit bounds for the diameter of word norms on [Formula: see text] given by conjugacy classes thereby continuing an investigation into such diameters by Kedra et al. Additionally, we present some examples of [Formula: see text]-pseudo-good rings and classify normally generating subsets of [Formula: see text] for [Formula: see text] the ring of algebraic integers in any number field.