On the bounded generation of arithmetic SL2

Let K be a number field and S be a finite set of primes of K containing the archimedean valuations. Let 𝒪 be the ring of S-integers in K. Morgan, Rapinchuck, and Sury [A. V. Morgan et al., Algebra Number Theory 12, 1949–1974 (2018)] have proved that if the group of units 𝒪× is infinite, then every matrix in SL2(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that K has at least 1 real embedding or S contains a finite place we can get a product of at most 8 elementary matrices. If we assume a suitable generalized Riemann hypothesis, then every matrix in SL2(𝒪) is the product of at most 5 elementary matrices if K has at least 1 real embedding, the product of at most 6 elementary matrices if S contains a finite place, and the product of at most 7 elementary matrices in general.

Bounded generation of wreath products

We show that the (standard restricted) wreath product of groups is boundedly generated if and only if the bottom group is boundedly generated and the top group is finite.