Embedding arithmetic hyperbolic manifolds

Abstract

This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov-Pyatetski-Shapiro and Agol-Belolipetsky-Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume ≤ v that bounds geometrically is at least v Cv , for v large enough.A. K. and S. R. were supported by the SNSF project no.

Counting cusped hyperbolic 3-manifolds that bound geometrically

“…Combining the pioneering result by Long and Reid [22] with the recent work [20] by Reid, Slavich and the first author, many arithmetic hyperbolic manifolds (in arbitrary dimension, compact or noncompact) bound geometrically. The proof there relies on arithmetic techniques and does not provide any counting results on the number of such manifolds with bounded volume.…”

Many cusped hyperbolic 3-manifolds do not bound geometrically

“…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 . We remark that many arithmetic hyperbolic 3-manifolds of simplest type embed geodesically by [15].…”

The Trace Field of Hyperbolic Gluings

“…This is [BHW11, Corollary 1.12]. This can be used to construct arithmetic manifolds that contain a hypersurface isometric to a fixed arithmetic manifold, see [KRS18]. For us, it will be useful in Section 4.5.2 when we consider specific examples of manifolds obtained by "closing up" arithmetic pieces.…”