Counting cusped hyperbolic 3-manifolds that bound geometrically

Abstract

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.

Embedding arithmetic hyperbolic manifolds

“…In addition to the above list, at the moment we do not know if one of β 3 (v) or B 3 (v) is finite for v sufficiently large (c.f. [19,Question 1.6]). Recall that β 3 (v) denotes the number of 3-dimensional hyperbolic geometric boundaries of volume ≤ v up to isometry, and B 3 (v) is the number of commensurability classes of 3-dimensional hyperbolic geometric boundaries of volume ≤ v.…”

“…The geometric boundaries constructed in [10] are arithmetic. The same lower bound is provided for the number of non-arithmetic 3-manifolds that bound geometrically, and for the number of 4-manifolds with connected geodesic boundary by virtue of an explicit construction [19].…”

“…The manifolds which are shown to embed geodesically in those papers are arithmetic. Some non-arithmetic 3manifolds which embed geodesically are produced in [19] by means of a rightangled hyperbolic 4-polytope. In this paper we show that many non-arithmetic manifolds of arbitrary dimension embed geodesically.…”

Many cusped hyperbolic 3-manifolds do not bound geometrically

“…A hyperbolic n-manifold M bounds geometrically if it is isometric to ∂W , for a hyperbolic (n + 1)-manifold W with totally geodesic boundary, c.f. [18], and also [14,16,19,20,22,28,29] for further progress in this topic. A hyperbolic n-manifold M is said to embed geodesically if there exists a hyperbolic (n + 1)-manifold N that contains a totally geodesic hypersurface isometric to M .…”