Mathematical Research Letters

Volume 30 (2023)

Number 2

Existence of an exotic plane in an acylindrical 3-manifold

Pages: 611 – 631



Yongquan Zhang (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.; and Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York, U.S.A.)


Let $P$ be a geodesic plane in a convex cocompact, acylindrical hyperbolic $3$-manifold $M$. Assume that $P^\ast = M^\ast \cap P$ is nonempty, where $M^\ast$ is the interior of the convex core of $M$. Does this condition imply that $P$ is either closed or dense in $M$? A positive answer would furnish an analogue of Ratner’s theorem in the infinite volume setting.

In [$\href{}{9}$] it is shown that $P^\ast$ is either closed or dense in $M^\ast$. Moreover, there are at most countably many planes with $P^\ast$ closed, and in all previously known examples, $P$ was also closed in $M$.

In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair $(M, P)$ such that $P^\ast$ is closed in $M^\ast$ but $P$ is not closed in $M$. In particular, the answer to the question above is no. Thus Ratner’s theorem fails to generalize to planes in acylindrical $3$-manifolds, without additional restrictions.

Received 1 March 2021

Received revised 29 July 2021

Accepted 3 January 2022

Published 13 September 2023