Small intersection numbers in the curve graph (Q2922864)

Let \(S=S_{g,p}\) denote an orientable surface of genus \(g\geq 0\) with \(p\geq 0\) punctures and \(\omega(S) = 3g + p - 3 > 1\). The curve graph \(C_{1}(S)\) for \(S\) is the graph whose vertices correspond to the isotopy classes of essential, non-peripheral simple closed curves on \(S\), and whose edges join vertices that represent curves whose union is a 2-component multi-curve. Denote the distance in this graph by \(d_{S}\). The curve graph was introduced by \textit{W. J. Harvey} [Ann. Math. Stud. 97, 245--251 (1981; Zbl 0461.30036)] and has since become a central tool to understand the mapping class group of a surface and hyperbolic structures on surfaces and \(3\)-manifolds. NEWLINENEWLINENEWLINENEWLINE In this note, the authors construct explicit curves (that have exactly the distance \(k\) for \(k \geq 4\)) that are as simple as possible in terms of their geometric intersection number on the surface. It is known that given a pair of curves \(\alpha, \beta\) on \(S\) one hasNEWLINENEWLINENEWLINE\[NEWLINE d_{S}(\alpha, \beta) \leq 2 \log_{2}(i(\alpha, \beta)) + 2. NEWLINE\]NEWLINE NEWLINENEWLINEAlso the purpose of this note is to establish a corresponding upper bound for the minimal number of times that a pair of distance \(k\) simple closed curves intersect. NEWLINENEWLINENEWLINENEWLINE Theorem 1.1. For any \(g, p\) with \(\omega = \omega(g, p)>1\), there exists an infinite geodesic ray \(\gamma =(v_{0},v_{1},v_{2},\dots)\) in \(C_{1}(S_{g,p})\) such that for any \(i\leq j\) NEWLINENEWLINE\[NEWLINE i(v_{i}, v_{j}) \leq \epsilon(\epsilon B)^{2j-5} \omega^{|j-i|-2} + f_{i,j}(\omega), NEWLINE\]NEWLINE where \(f_{i,j}(x)\) is \(O(x^{j-i-4})\), \(B\) is a universal constant, and \(\epsilon = 1\) if \(g\geq 2\) and \(\epsilon = 4\) otherwise.NEWLINENEWLINEThis theorem was proved in response to the following question (Dan Margarit): Is it true that for fixed \(k\) the function \(i_{k,g}\) is \(O(g^{k-2})\)?