Algebraic intersection in regular polygons

Abstract

We study the function $$\mbox{KVol} : (X,ω)\mapsto \mbox{Vol} (X,ω) \sup_{α,β} \frac{\mbox{Int} (α,β)}{l_g (α) l_g (β)}$$ defined on the moduli spaces of translation surfaces. More precisely, let ${\mathcal{T}}_n$ be the Teichmüller discs of the original Veech surface $(X_n,{\omega }_n)$ arising from right-angled triangle with angles $(\pi /2,\pi /n,(n-2)\pi /2n)$ by the unfolding construction for $n\ge 5$. For $n\equiv 1\mathrm{mod}2$ and any $(X,\omega )\in {\mathcal{T}}_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \fracπ{n} \leq \mbox{KVol}(X,ω) \leq \frac{n}{2} \cot \fracπ{n} \cdot \frac1{\sin \frac{2π}{n}}.$$ The lower bound is uniquely realized at $(X_n,{\omega }_n)$.

New version with the first author added and completely different methods. We focus on the $n=2m+1$ case, the $n=4m$ case is dealt with in a forthcoming paper by the first author. 30pages, 15 figures