  Let $M$ be a $\rm II_1$ factor and let $\mathcal{F}(M)$ denote the
fundamental group of $M$. In this article, we study the following property of
$M$: for arbitrary $\rm II_1$ factor $B$, we have $\mathcal{F}(M
\overline{\otimes} B)=\mathcal{F}(M)\mathcal{F}(B)$. We prove that for any
subgroup $G\leq \mathbb{R}^*_+$ which is realized as a fundamental group of a
$\rm II_1$ factor, there exists a $\rm II_1$ factor $M$ which satisfies this
property and whose fundamental group is $G$. Using this, we deduce that if $G,H
\leq \mathbb{R}^*_+$ are realized as fundamental groups of $\rm II_1$ factors
(with separable predual), then so are groups $G \cdot H$ and $G \cap H$.
