Ascending chains of irreducible lattices, bi-reversible automata and affine arithmetic groups

For each $n\ge 2$, we construct an ascending chain of irreducible lattices in the product of $n$ homogeneous trees. Moreover, for each pair of integers $m_1,m_2\ge 1$, we define explicitly a bi-reversible automaton $\mathcal{B}$ such that the group $G_{\mathcal{B}}$ defined by the automaton $\mathcal{B}$ has finiteness length $m_1$ (i.e. it is of type ${\mathrm{F}}_{m_1}$ but not of type ${\mathrm{F}\mathrm{P}}_{m_1+1}$), and the group $G_{{\mathcal{B}}^{\ast }}$ defined by the dual automaton has finiteness length $m_2$. Both constructions rely on the consideration of $S$-arithmetic groups in the affine group of a global function field.