We introduce a class of countable groups by some abstract group-theoretic conditions. This class includes linear groups with finite amenable radical and finitely generated residually finite groups with some nonvanishing ${\ell }^2$-Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group $\Gamma$ in this class, we determine the general structure of the possible lattice embeddings of $\Gamma$, that is, of all compactly generated, locally compact groups that contain $\Gamma$ as a lattice. This leads to a precise description of possible nonuniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.

We investigate the analogue of the quantum unique ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the generalized Riemann hypothesis (GRH), we establish both QUE for half-integral weight Hecke Maaß cusp forms for ${\Gamma }_0\left(4\right)$ lying in Kohnen’s plus subspace and mass equidistribution for half-integral weight holomorphic Hecke cusp forms for ${\Gamma }_0\left(4\right)$ lying in Kohnen’s plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp forms equidistribute with respect to hyperbolic measure on ${\Gamma }_0\left(4\right)\backslash \mathbb{H}$ as the weight tends to infinity.

We provide a description of the sheaves of Kähler differentials of the arc space and jet schemes of an arbitrary scheme where these sheaves are computed directly from the sheaf of differentials of the given scheme. Several applications on the structure of arc spaces are presented.