# find a special free subsemigroup

It is well-known theorem that for an elementary amenable group $G$, $G$ has exponential growth rate iff $G$ contains non cyclic free semigroup.

Now I am interested in the following questions:

Let $G=\mathbb{Z}^2\rtimes_{\sigma}\mathbb{Z}=\langle y,z,x|yz=zy, xyx^{-1}=y^2z, xzx^{-1}=yz\rangle$, which is a polycyclic group and has exponential growth rate, so by the above theorem, we can find a pair $(a,b)\in G$ such that $a, b$ generates a free subsemigroup.

1, Are there any concrete examples of these $(a,b)$ in the above group?

2, Especially, can we find such one pair $(a,b)$ with the following additional property:

For any $n\geq 1$ and any $g_1,\cdots, g_n\in \langle a,b\rangle,$ we can find some $g\in \langle a,b\rangle$, such that $gg_i\in\langle a,b\rangle^{+},\forall 1\leq i\leq n$.

Here, $\langle a,b\rangle$ denotes the subgroup generated by $a,b$, and $\langle a,b\rangle^{+}$ denotes the free semigroup generated by $a,b$.

3, If the above question has a negative answer, could anyone give me a polycyclic group $G$ where the above answer is positive?

For 1: yes there are concrete free subsemigroups: if you write the matrix $\sigma=\begin{pmatrix} 2 & 1 \\ 1 & 1\end{pmatrix}$, then your group embeds into $\mathbf{R}^2\rtimes_\sigma\mathbf{Z}$, and $\sigma$ is $\mathbf{R}$-diagonalizable. Hence by projecting modulo a real eigenline, you get a dense embedding into $\mathbf{R}\rtimes_t\mathbf{Z}$, where the action is by multiplication by $t$ (one of the eigenvalues of $\sigma$). Of course the latter embeds into the affine group $\mathbf{R}\rtimes\mathbf{R}$.