Volume growth in the component of fibered twists

TL;DR

This paper studies the volume growth of fibered twists in Liouville domains, providing lower bounds and conditions for infinite order, with applications to singularities and symplectic hypersurfaces.

Contribution

It introduces a uniform lower bound for volume growth using wrapped Floer homology and establishes infinite order of fibered twists under certain conditions.

Findings

Lower bounds for volume growth of fibered twists.

Infinite order of fibered twists in certain cases.

Explicit computations of wrapped Floer homology for specific examples.

Abstract

For a Liouville domain $W$ whose boundary admits a periodic Reeb flow, we can consider the connected component $[\tau] \in \pi_0(\text{Symp}^c(\widehat W))$ of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, of the component $[\tau]$ and give a uniform lower bound of the growth using wrapped Floer homology. We also show that $[\tau]$ has infinite order in $\pi_0(\text{Symp}^c(\widehat W))$ if there is an admissible Lagrangian $L$ in $W$ whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of $A_k$-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse-Bott spectral sequences.