Let $G=\text{SL}(3,\mathbb R)$ and $\Gamma=\text{SL}(3,\mathbb Z)$. Let $a(\lambda)$ be the diagonal matrix $a(\lambda):=\text{diag}(\lambda,1,\lambda^{-1}),$ where $\lambda>0$.
Take $x\in G/\Gamma$ and consider $\Phi:=\{g\in G:gx=x\}$, where the action is just the usual left action on the coset space.
Let $V_1=\{v_1(t):t\in \mathbb R\}$, where $v_1(t)$ is the matrix
$$v_1(t):=\begin{bmatrix} 1 & t & t^2/2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{bmatrix}.$$
I wonder if the product set $V_1 a(\lambda) \Phi$ is a closed subset of $G$. (Maybe the fixed element $a(\lambda)$ is irrelevant here, but I am not very sure about that. )