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FRACTIONAL POWERS OF DEHN TWISTS ABOUT NONSEPARATING CURVES

Published online by Cambridge University Press:  02 September 2013

KASHYAP RAJEEVSARATHY
KASHYAP RAJEEVSARATHY*
Affiliation:
Department of Mathematics, Indian Institute of Science Education Research Bhopal, ITI (Gas Rahat) Building, Govindpura, Bhopal 462023, Madhya Pradesh, India e-mail: [email protected]
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Abstract

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Let 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.

Keywords

57M9957M60
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 1 , January 2014 , pp. 197 - 210
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Harvey, W. J., Cyclic groups of automorphisms of a compact Riemann surface, Q. J. Math. 17 (2) (1966), 86–97.Google Scholar
2.Margalit, D. and Schleimer, S., Dehn twists have roots, Geom. Topol. 13 (3) (2009), 1495–1497.CrossRefGoogle Scholar
3.McCullough, D. and Rajeevsarathy, K., Roots of Dehn twists, Geom. Dedicata 151 (2011), 397–409.CrossRefGoogle Scholar
4.Rajeevsarathy, K., GAP software for computing SE data sets, available at: home.iiserb.ac.in/~kashyap/fracpowerse.g.Google Scholar
5.Rajeevsarathy, K., GAP software for computing SP data sets, available at: home.iiserb.ac.in/~kashyap/fracpowersp.g.Google Scholar
6.Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc. 15 (5) (1983), 401–487.CrossRefGoogle Scholar
7.Thurston, W. P., The geometry and topology of three-manifolds, notes available at: http://www.msri.org/communications/books/gt3m/PDF.Google Scholar
8.Wiman, A., Ueber die hyperelliptischen curven and diejenigen vom geschlechte p = 3, welche eindeutigen transformationen in sich zulassen, Bihang Kongl. Svenska Vetenskaps-Akademiens Handilgar (Stockholm 1895–6).Google Scholar