The natural flow and the critical exponent

TL;DR

The paper introduces the natural flow on non-positively curved manifolds, linking dynamical and geometric invariants to the critical exponent, and applies it to inequalities, conjecture resolutions, and dimension bounds.

Contribution

It constructs the natural flow and demonstrates its applications in relating invariants, resolving conjectures, and establishing bounds on geometric and topological properties.

Findings

Established $k$-dimensional isoperimetric inequalities for $k > \delta$

Resolved a conjecture on the non-existence of certain complex subvarieties

Provided bounds on homological and cohomological dimensions

Abstract

Inspired by work of Besson-Courtois-Gallot, we construct a flow called the natural flow on a non-positively curved Riemannian manifold $M$. As with the natural map, the $k$-Jacobian of the natural flow is directly related to the critical exponent $\delta$ of the fundamental group. There are several applications of the natural flow that connect dynamical, geometrical, and topological invariants of the manifold. First, we give $k$-dimensional linear isoperimetric inequalities when $k > \delta$. This, in turn, produces lower bounds on the Cheeger constant. We resolve a recent conjecture of Dey-Kapovich on the non-existence of $k$-dimensional compact, complex subvarieties of complex hyperbolic manifolds with $2k > \delta$. We also provide upper bounds on the homological dimension, generalizing work of Kapovich and work of Farb with the first two authors. Using the natural flow together with Morse theory, we also give upper bounds on the cohomological dimension, which partially resolve a conjecture of Kapovich. Finally, we introduce a new growth condition on the Bowen-Margulis measure that we call uniformly exponentially bounded that we connect to the cohomological dimension and which could be of independent interest.