Let $M = SL_2(\mathbb{Z}[i])\backslash \mathbb{H}^{(3)}$ be the Bianchi orbifold, and let $\{f_n\}_{n=1}^\infty \subset L^2(M)$ be a sequence of Hecke--Maass forms on it. We should that the probability measures on $M$ with densities $|f_n(x)|^2$ with respect to the Riemannian volume become equidistributed on $M$. This generalizes the case $SL_2(\mathbb{Z})\backslash \mathbb{H}^{(2)}$ due to Lindenstrauss.
In the first part of the talk I will introduce the quantum unique ergodicity problem and the setup necessary to state the theorem. In the second half I will try to give ideas of the proof, including of the full result. More generally the result holds when we replace $\mathbb{Q}(i)$ with a general number field $F$, $SL_2$ with the group $G$ of norm-1 elements of a quaternion algebra over $F$, $\mathbb{H}^{(3)}$ with $G(F_\infty)/K_\infty$ where $F_\infty = F\otimes_\mathbb{Q} \mathbb{R}$ and $K_\infty$ is a maximal compact subgrouop, and $SL_2(\mathbb{Z}[i])$ with a congruence lattice.
Joint work with Z. Shem-Tov, also relying on work with A. Zaman.