Mathematical Research Letters

Volume 29 (2022)

Number 1

Embedding non-arithmetic hyperbolic manifolds

Pages: 247 – 274



Alexander Kolpakov (Institut de Mathématiques, Université de Neuchâtel, Switzerland)

Stefano Riolo (Dipartimento di Matematica, Università di Pisa, Italy)

Leone Slavich (Dipartimento di Matematica “F. Casorati”, Università di Pavia, Italy)


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 $\leq v$ that bounds geometrically is at least $v^{Cv}$, for $v$ large enough.

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A.K. and S.R. were supported by the SNSF project no. PP00P2-170560.

Received 6 May 2020

Accepted 11 October 2020

Published 6 September 2022