• hyperbolic manifolds
• RFRS
• virtual fibrations

The virtual fibering conjecture was first stated by Thurston in 1982. He asked if every hyperbolic 3-manifold admits a finite cover which fibers over the circle. It has been proved only in the last decade, but finding such a cover is still an open problem.
In this work we describe a criterion for an arithmetic hyperbolic lattice to admit a RFRS tower consisting entirely of congruence subgroups.
In the first chapter of the thesis we recall basic notions of hyperbolic geometry and we present some geometric background about the virtual fibration problem.
The second chapter treats algebraic notions needed for the main construction, such as number fields, non-Archimedean valuations, $\mathfrak{p}$-adic completions and especially buildings. We define the Bruhat-Tits building associated to some groups of matrices over the field of p-adic rational numbers $\mathbb{Q}_p$, and we present some of their properties.
Finally, in the last chapter we describe the criterion and we show the explicit construction in two examples. In the first one we define a congruence RFRS tower for the fundamental group of the Magic manifold, using the Bruhat-Tits tree associated to $SL(2,\mathbb{Q}_2)$. In the second example we use the Bruhat-Tits tree of $O(4,1;\mathbb{Q}_2)$ in order to define a congruence RFRS tower for the level-4 congruence subgroup of the group $O(4,1;\mathbb{Z})$. By embedding, up to commensurability, the Bianchi groups in $O(4,1;\mathbb{Z})$ we prove the main result of this work.