Enumerating pseudo-Anosov foliations

A. Papadopoulos and R. C. Penner

Vol. 142 (1990), No. 1, 159–173

DOI: 10.2140/pjm.1990.142.159

Abstract

Let M be a closed oriented surface of genus at least two, and let 𝒫ℱ denote the space of all projective classes of measured foliations on M. The authors have previously given a criterion in terms of certain combinatorial words for an element of 𝒫ℱ to be left invariant by a pseudo-Anosov map of M: such foliations are characterized by the fact that the associated word is eventually periodic. The current work derives an estimate which says roughly that the dilatation of the corresponding pseudo-Anosov map is large if the periodic part of the word is long. This estimate is then used to bound the number of distinct conjugacy classes of foliations invariant under pseudo-Anosov maps of M in terms of a specified bound on the dilatations.

Mathematical Subject Classification 2000

Primary: 57M99

Secondary: 57R30, 58F18

Milestones

Received: 10 March 1987

Revised: 31 October 1988

Published: 1 March 1990