Hofer geometry of $A_3$-configurations

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 $\mathcal{L}(L_0)$ and $\mathcal{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $({\mathbb{R}}^{\mathrm{\infty }},\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty }})$, 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 $CF({\tau }^{2\mathrm{\ell }}(L_0),L^{\mathrm{\prime }})$ is unbounded in $L^{\mathrm{\prime }}\in \mathcal{L}(L_2)$ for any $\mathrm{\ell }\in {\mathbb{N}}_0$.