Hofer geometry of A₃-configurations

Let L₀,L₁,L₂ ⊂ M be exact Lagrangian spheres in a Liouville domain M with 2c₁(M)=0. If L₀,L₁,L₂ are in an A₃-configuration, we show that L(L₀) and L(L₂) when endowed with the Hofer metric contain quasi-flats of arbitrary dimension. We then strengthen this result to show that L(L₀) and L(L₂) both contain quasi-isometric embeddings of (R^∞, d_∞), i.e. infinite-dimensional quasi-flats. We also obtain as a corollary of the proof that Hamc(M) contains an infinite-dimensional quasi-flat if M contains an A₃-configuration of exact Lagrangian spheres. Additionally, we show the same for the two zero-sections in an A₂-plumbing. Along the way, we obtain the same result for L(F), where F is any fiber in the cotangent bundle of a sphere Sⁿ. This gives an alternative proof of a result of Usher for n ≥ 3 and Feng-Zhang for $n=2$. Lastly, we use our method to show that for a Dehn twist τ: M → M along L₁ the boundary depth of HF(τ⁽L₀), L') is unbounded in L' ∈ L(L₂) for any ∈ 2N₀.