Hofer geometry of $A_3$-configurations

Abstract

Let $L_0,L_1,L_2 \subset M$ be exact Lagrangian spheres in a Liouville domain $M$ with $2c_1(M)=0$. If $L_0,L_1,L_2$ form an $A_3$-configuration, we show that $\mathscr{L}(L_0)$ and $\mathscr{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $(\mathbb{R}^\infty, \|\cdot\|_\infty)$, i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that $\text{Ham}_c(M)$ itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist $\tau: M \to M$ along $L_1$ the boundary depth of $CF(\tau^{2\ell}(L_0), L')$ is unbounded in $L' \in \mathscr{L}(L_2)$ for any $\ell \in \mathbb{N}_0$.