Bounded generation for congruence subgroups of ${\mathrm{S}\mathrm{p}}_4(R)$

This paper describes a bounded generation result concerning the minimal natural number $K$ such that for $Q(C_2,2R):={A\varepsilon_{\phi}(2x)A^{-1}|x\in R,A\in{\rm Sp}4®,\phi\in C_2}$,onehas$N{C_2,2R}={X_1\cdots X_K|\forall 1\leq i\leq K:X_i\in Q(C_2,2R)}$forringsofalgebraicintegers$R$ and the principal congruence subgroup $N_{C_2,2R}$ in ${\mathrm{S}\mathrm{p}}_4(R)\mathrm{.}$ 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 ${\mathrm{S}\mathrm{p}}_4(R)$ 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 ${\mathrm{S}\mathrm{p}}_4(R)$ for $R$ a ring of algebraic integers.