## Geometry & Topology

### Right-veering diffeomorphisms of compact surfaces with boundary II

#### Abstract

We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [Invent. Math. 169 (2007) 427–449]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group $Bn$ on $n$ strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Proc. London Math. Soc. (3) 82/83 (2001) 747–768/443–471].

#### Article information

Source
Geom. Topol., Volume 12, Number 4 (2008), 2057-2094.

Dates
Revised: 22 April 2008
Accepted: 18 June 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800122

Digital Object Identifier
doi:10.2140/gt.2008.12.2057

Mathematical Reviews number (MathSciNet)
MR2431016

Zentralblatt MATH identifier
1170.57013

#### Citation

Honda, Ko; Kazez, William H; Matić, Gordana. Right-veering diffeomorphisms of compact surfaces with boundary II. Geom. Topol. 12 (2008), no. 4, 2057--2094. doi:10.2140/gt.2008.12.2057. https://projecteuclid.org/euclid.gt/1513800122

#### References

• J Barge, É Ghys, Cocycles d'Euler et de Maslov, Math. Ann. 294 (1992) 235–265
• Y M Eliashberg, W P Thurston, Confoliations, University Lecture Series 13, Amer. Math. Soc. (1998)
• J B Etnyre, Legendrian and transversal knots, from: “Handbook of knot theory”, Elsevier B. V., Amsterdam (2005) 105–185
• J-M Gambaudo, É Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems 24 (2004) 1591–1617
• J-M Gambaudo, É Ghys, Braids and signatures, Bull. Soc. Math. France 133 (2005) 541–579
• É Ghys, Groups acting on the circle, Enseign. Math. $(2)$ 47 (2001) 329–407
• E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: “Proc. of the ICM, Vol. II (Beijing, 2002)”, Higher Ed. Press, Beijing (2002) 405–414
• A Hatcher, Some examples of essential laminations in $3$–manifolds, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 313–325
• K Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000) 309–368
• K Honda, W H Kazez, G Matić, On the contact class in Heegaard Floer homology
• K Honda, W H Kazez, G Matić, Tight contact structures on fibered hyperbolic $3$–manifolds, J. Differential Geom. 64 (2003) 305–358
• K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007) 427–449
• S Y Orevkov, Markov moves for quasipositive braids, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 557–562
• J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844
• R Roberts, Taut foliations in punctured surface bundles. I, Proc. London Math. Soc. $(3)$ 82 (2001) 747–768
• R Roberts, Taut foliations in punctured surface bundles. II, Proc. London Math. Soc. $(3)$ 83 (2001) 443–471
• W P Thurston, H E Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975) 345–347