  Let $X \subset \mathbb{C}^n$ be an algebraic variety, and let $\Lambda
\subset \mathbb{C}^n$ be a discrete subgroup whose real and complex spans
agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^n
/ \Lambda$, thereby extending a result of Peterzil-Starchenko in the case when
$\Lambda$ is cocompact.
  We also obtain a similar extension when $X\subset \mathbb{R}^n$ is definable
in an o-minimal structure with no restrictions on $\Lambda$, and as an
application prove the following conjecture of Gallinaro: for a closed
semi-algebraic $X\subset \mathbb{C}^n$ (such as a complex algebraic variety)
and $\exp:\mathbb{C}^n\to (\mathbb{C}^*)^n$ the coordinate-wise exponential
map, we have $\overline{\exp(X)}=\exp(X)\cup \bigcup_{i=1}^m \exp(C_i)\cdot
\mathbb{T}_i$ where $\mathbb{T}_i\subset (\mathbb{C}^*)^n$ are
positive-dimensional compact real tori and $C_i\subset \mathbb{C}^n$ are
semi-algebraic.
