Minimally intersecting filling pairs on surfaces (Q2346727)

Let \(S_g\) denote the orientable closed surface of genus \(g\), and Mod\((S_g)\) the mapping class group of \(S_g\). A pair \((\alpha, \beta)\) of simple closed curves is said to fill \(S_g\) if \(S_g \setminus (\alpha \cup \beta)\) is a disjoint union of disks. Let \(i (\alpha, \beta)\) denote the geometric intersection number of \((\alpha, \beta)\), then \(i(\alpha, \beta) \geq 2g-1\) when \(g \not= 2\), and \(i(\alpha, \beta) \geq 4\) when \(g = 2\). Therefore, a filling pair \((\alpha, \beta)\) is called a minimally intersecting filling pair if \(i(\alpha, \beta) = 2g-1\) when \(g \not= 2\), and \(i(\alpha,\beta)=4\) when \(g=2\). Let \(N(S_g)\) denote the number of Mod\((S_g)\)-orbits of minimally intersecting filling pairs. It is shown that (Theorem 1.1) \(f(g) \leq N(g) \leq 2^{2g-2}(4g-5)(2g-3)!\), where \(f(g)\) satisfies \(0 < \lim_{g \to \infty} g^2 f(g) / 3 ^{g/2} < \infty\). For a filling pair \((\alpha, \beta)\) on \(S_g\), \((g > 2)\), define \(T_1(\alpha,\beta)\) to be the number of simple closed curves intersecting \(\alpha \cup \beta\) only once. It is shown that (Theorem 1.2) \(T_1(\alpha, \beta) \leq 4g-2\) with equality if \((\alpha, \beta)\) is minimally intersecting. Let \(\mathcal{F}_g : \mathcal{M}(S_g) \to \mathbb{R}\) be the map from the moduli space \(\mathcal{M}(S_g)\) of Riemann surfaces of genus \(g\) to \(\mathbb{R}\) defined by \(\mathcal{F}_g(\sigma)\) for \(\sigma \in \mathcal{M}(S_g)\) to be the length of the shortest minimally intersecting filling pair with respect to \(\sigma\). It is shown that (Theorem 1.3) \(\mathcal{F}_g\) is proper and a topological Morse function such that, for any \(\sigma \in \mathcal{M}(S_g)\), \(\mathcal{F}_g(\sigma) \geq m_g/2\), where \(m_g\) denotes the perimeter of a regular, right-angled \((8g-4)\)-gon. Let \(\mathcal{B}_g\) be the set of points in \(\mathcal{M}(S_g)\) whose value of \(\mathcal{F}_g\) is \(m_g/2\). Then the number of \(\mathcal{B}_g\) is finite and grows at least exponentially in \(g\), and, for each \(\sigma \in \mathcal{B}_g\), the injectivity radius of \(\sigma\) is at least \((\cosh^{-1} (9/\sqrt{73}))/2\).