Bounded generation for congruence subgroups of ${\rm Sp}_4(R)$

Abstract

This paper describes a bounded generation result concerning the minimal natural number $K$ such that for $Q(C_2,2R):=\{A\varepsilon_{\phi}(2x)A^{-1}|x\in R,A\in{\rm Sp}_4(R),\phi\in C_2\}$, one has $N_{C_2,2R}=\{X_1\cdots X_K|\forall 1\leq i\leq K:X_i\in Q(C_2,2R)\}$ for rings of algebraic integers $R$ and the principal congruence subgroup $N_{C_2,2R}$ in ${\rm Sp}_4(R).$ This gives an explicit version of an abstract bounded generation result of a similar type as presented by Morris. Furthermore, the result presented does not depend on several number-theoretic quantities unlike Morris' result. Using this bounded generation result, we further give explicit bounds for the strong boundedness of ${\rm Sp}_4(R)$ for certain examples of rings $R,$ thereby giving explicit versions of results in an earlier paper. We further give a classification of normally generating subsets of ${\rm Sp}_4(R)$ for $R$ a ring of algebraic integers.