ROOTS OF DEHN TWISTS

Abstract

Let C be a curve in a closed orientable surface F of genus g &ge 2 that separates F of genera gi, for i = 1,2. We study the set of roots in Mod(F) of the Dehn twist tC about C. All roots arise from pairs of Cni actions on the subsurfaces of genera gi, where n=lcm(n1,n2) is the degree of the root, that satisfy a certain compatibility condition. The Cni actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, allowing a classification of all roots of tC by compatible pairs of data sets. We use these data set pairs to classify all roots for g = 2 and g = 3. We show that there is always a root of degree at least 2g2+2g, while n &le 4g2+2g. We also give some additional applications.