Small filling sets of curves on a surface (Q616946)

The authors study the size of a collection of simple closed curves on a closed orientable surface of genus \(g \geqslant 2\) which fill the surface and pairwise intersect at most \(K\) times for some fixed \(K \geqslant 1\). Denote by \(n\) the number of curves in the collection. Theorem 1 states that \(n\) satisfies \(n^2 - n \geqslant (4g - 2)/K\). Moreover, if \(N\) is the smallest integer satisfying this inequality, then there exists a set of no more than \(N+1\) such curves on \(S\). As a corollary, \(n \sim 2 \sqrt{g}/\sqrt{K}\) as \(g \to \infty\). At the same time, the number of systoles in a smallest filling set grows with \(g\) at a rate of order strictly greater than \(\sqrt{g}\). More precisely, Theorem 2 states the following. Let \(S\) be a closed, orientable hyperbolic surfaces of genus \(g \geqslant 2\) with a filling set of systoles \(\{ \sigma_{1}, \dots, \sigma_{n}\}\). Then \(n \geqslant \pi \sqrt{g(g-1)} / \log (4g-2)\). Furthermore, there exist hyperbolic surfaces of genus \(g\) with filling sets of \(n \leqslant 2g\) systoles. The obtained results illustrate that the topological condition that curves in a set of curves pairwise intersect at most once is quite far from the geometric condition that such a set of curves can arise as systoles.