The normal growth exponent of a codimension-1 hypersurface of a negatively curved manifold

TL;DR

This paper investigates the geometric and algebraic implications of the normal growth exponent of a codimension-1 hypersurface in negatively curved manifolds, establishing conditions under which the ambient space is hyperbolic and its fundamental group is a lattice.

Contribution

It introduces the concept of the normal growth exponent, proves that a bound of 1 implies the ambient space is hyperbolic, and demonstrates the necessity of this bound in higher dimensions.

Findings

If the normal growth exponent is at most 1, the ambient manifold is bi-Lipschitz to hyperbolic space.

A totally geodesic hypersurface with controlled normal growth exponent implies the ambient space's hyperbolicity.

The bound on the normal growth exponent is necessary in dimensions at least 4.

Abstract

Let $X$ be a Hadamard manifold with pinched negative curvature $-b^2\leq\kappa\leq -1$. Suppose $\Sigma\subseteq X$ is a totally geodesic, codimension-1 submanifold and consider the geodesic flow $\Phi^\nu_t$ on $X$ generated by a unit normal vector field $\nu$ on $\Sigma$. We say the normal growth exponent of $\Sigma$ in $X$ is at most $\beta$ if \[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d \Phi_t^\nu \Vert_{\infty} }{ e^{\beta \vert t \vert}} < \infty, \] where $\Vert d \Phi_t^\nu \Vert_{\infty} $ is the supremum of the operator norm of $d \Phi_t^\nu $ over all points of $\Sigma$. We show that if $\Sigma$ is bi-Lipschitz to hyperbolic $n$-space $\mathbb{H}^n$ and the normal growth exponent is at most 1, then $X$ is bi-Lipschitz to $\mathbb{H}^{n+1}$. As an application, we prove that if $M$ is a closed, negatively curved $(n+1)$-manifold, and $N\subset M$ is a totally geodesic, codimension-1 submanifold that is bi-Lipschitz to a hyperbolic manifold and whose normal growth exponent is at most 1, then $\pi_1(M)$ is isomorphic to a lattice in $\text{Isom}(\mathbb{H}^{n+1})$. Finally, we show that the assumption on the normal growth exponent is necessary in dimensions at least 4.