We introduce and study some deformations of complete finitevolume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling.We construct in particular an analytic path of complete, finite-volume cone four-manifolds Mt that interpolates between two hyperbolic four-manifolds M 0 and M 1 with the same volume 8 3 π 2 . The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from 0 to 2π. Here, the singularity of Mt is an immersed geodesic surface whose cone angles also vary monotonically from 0 to 2π. When a cone angle tends to 0 a small core surface (a torus or Klein bottle) is drilled producing a new cusp.We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to 2π, like in the famous figure-eight knot complement example.The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm. 1 arXiv:1608.08309v4 [math.GT] 15 Sep 2017 Area(T ) = 4π − 2β, Area(K) = 4π − 2α.When t varies from 0 to 1 the angle α goes from 0 to 2π and β goes from 2π to 0.The path converges as t → 0 and t → 1 to two complete, finite-volume hyperbolic four-manifolds M 0 = int(M ) \ T and M 1 = int(M ) \ K.Structure of the paper. The paper is organized as follows. In Section 2 we recall some well-known facts about (acute-angled) polytopes, Coxeter diagrams,

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We prove that there are at least two commensurability classes of (cusped, arithmetic) minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold.

In this note, we show that there exist cusped hyperbolic 3-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds.

We show that the number of isometry classes of cusped hyperbolic 3-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and nonarithmetic settings.