FRACTIONAL POWERS OF DEHN TWISTS ABOUT NONSEPARATING CURVES

Abstract

AbstractLet Sg be a closed orientable surface of genus g ≥ 2 and C a simple closed nonseparating curve in F. Let tC denote a left-handed Dehn twist about C. A fractional power of tC of exponent ℓ//n is an h ∈ Mod(Sg) such that hn = tCℓ. Unlike a root of a tC, a fractional power h can exchange the sides of C. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if gcd(ℓ,n) = 1, then h will be isotopic to the ℓth power of an nth root of tC and that n ≤ 2g+1. In general, we show that n ≤ 4g, and that side-preserving fractional powers of exponents 2g//2g+2 and 2g//4g always exist. For a side-exchanging fractional power of exponent ℓ//2n, we show that 2n ≥ 2g+2, and that side-exchanging fractional powers of exponent 2g+2//4g+2 and 4g+1//4g+2 always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on S5.