  A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$
divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among
others, have shown that the gcd of two divisibility sequences corresponding to
subgroups of the multiplicative group grows in a controlled way. Silverman
conjectured that a similar behaviour should appear in many algebraic groups. We
extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and
prove an analogue of Silverman's conjecture over function fields for abelian
and split semiabelian varieties and some generalizations of this result. We
employ tools coming from the theory of unlikely intersections as well as
properties of the so-called Betti map associated to a section of an abelian
scheme.
