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: M M along L₁ the boundary depth of HF (^ (L₀), L') is unbounded in L' L (L₂) for any 2N₀.