(Joint work with P. Corvaja and D.
Masser.)
The topic of the talk arises from the
Manin-Mumford conjecture and its extensions, where we shall
focus on the case of (complex connected) commutative
algebraic groups $G$ of dimension $2$. The `Manin-Mumford'
context in these cases predicts finiteness for the set of
torsion points in an algebraic curve inside $G$, unless the
curve is of `special' type, i.e. a translate of an algebraic
subgroup of $G$.
In the talk we shall consider not merely the set of torsion
points, but its topological closure in $G$ (which turns out
to be also the maximal compact subgroup). In the case of
abelian varieties this closure is the whole space, but this is
not so for other $G$; actually, we shall prove that in certain
cases (where a natural dimensional condition is fulfilled) the
intersection of this larger set with a non-special curve
must still be a finite set.
We shall conclude by stating in brief some extensions of
this problem to higher dimensions.