Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces

Abstract

Let $S$ be a Riemann surface with a puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, $(a,f^m(a))$ fills $S$ for $m\ge 3$. We also study the cases of $0<m\le 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint. Then $(a,f^m(a))$ for any $m\ge 2$ fills $S$.

11 pages

References in corpus (1)

• The lower central series and pseudo-Anosov dilatations