## Geometry & Topology

### Area preserving group actions on surfaces

#### Abstract

Suppose $G$ is an almost simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group. For example any finite index subgroup of $SL(3,ℤ)$ is such a group. The main result of this paper is that every action of $G$ on a closed oriented surface by area preserving diffeomorphisms factors through a finite group.

#### Article information

Source
Geom. Topol., Volume 7, Number 2 (2003), 757-771.

Dates
Revised: 26 October 2003
Accepted: 29 October 2003
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883321

Digital Object Identifier
doi:10.2140/gt.2003.7.757

Mathematical Reviews number (MathSciNet)
MR2026546

Zentralblatt MATH identifier
1036.37010

#### Citation

Franks, John; Handel, Michael. Area preserving group actions on surfaces. Geom. Topol. 7 (2003), no. 2, 757--771. doi:10.2140/gt.2003.7.757. https://projecteuclid.org/euclid.gt/1513883321

#### References

• G Atkinson, Recurrence of cocycles and random walks, J. Lond. Math. Soc. 13 (1976) 486-488
• J Birman, A Lubotzky, J McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983) 1107-1120
• S Bleiler, A Casson, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, 9, Cambridge University Press (1988)
• M Brown, J Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984) 503–504
• C Conley, E Zehnder, The Birkhoff-Lewis Fixed point Theorem and a Conjecture of V.I. Arnold, Invent. Math. 73 (1983) 33–49
• B Farb, H Masur, Superrigidity and mapping class groups, Topology 37 (1998) 1169–1176
• B Farb, P Shalen, Real analytic actions of lattices, Invent. Math. 135 (1998) 271–296
• A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Asterisque 66–67 (1979)
• J Franks, Homology and Dynamical Systems, Amer. Math. Soc. CBMS Regional Conference Series 49 (1982), 120 pages.
• J Franks, Recurrence and fixed points of surface homeomorphisms, Ergod. Th. Dynam. Sys. 8* (1988) 99–107
• J Franks, Generalizations of the Poincaré-Birkhoff Theorem, Annals of Math. 128 (1988) 139–151
• J Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Inventiones Math. 108 (1992) 403–418
• J Franks, Area Preserving Homeomorphisms of Open Surfaces of Genus Zero, New York Jour. of Math. 2 (1996) 1–19
• J Franks, Rotation vectors and fixed points of area preserving surface diffeomorphisms, Trans. Amer. Math. Soc. 348 (1996) 2637–2662
• J Franks, Rotation numbers for Area Preserving Homeomorphisms of the Open Annulus, Proceedings of the International Conference Dynamical Systems and Related Topics, (K Shiraiwa, editor) World Scientific (1991) 123–128
• J Franks, M Handel, Periodic points of Hamiltonian surface diffeomorphisms, Geometry and Topology 7 (2003) 713–756
• M Handel, Zero entropy surface diffeomorphisms, preprint
• M Handel, A fixed point theorem for planar homeomorphisms, Topology 38 (1999) 235–264
• M Handel, Commuting homeomorphisms of $S^2$, Topology 31 (1992) 293–303
• M Handel, W P Thurston, New proofs of some results of Nielsen, Advances in Math. 56 (1985) 173–191
• G A Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17, Springer-Verlag, Berlin (1991)
• J Mather, Invariant subsets of area-preserving homeomorphisms of surfaces, Advances in Math. Suppl. Studies, 7B (1994) 331–351
• J Mather, Topological proofs of some purely topological consequences of Caratheodory's Theory of prime ends, in Selected Studies, (Th M Rassias, G M Rassias, editors) North-Holland (1982) 225–255
• L Polterovich, Growth of maps, distortion in groups and symplectic geometry.
• J Rebelo, On nilpotent groups of real analytic diffeomorphisms of the torus, C.R. Acad. Sci. Paris Sér. I Math. 331 (2000) 317-322
• J Tits, Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris Sér. A-B 283 (1976) Ai, A693–A695
• W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417–431
• D Witte, Arithmetic Groups of Higher $Q$-Rank Cannot Act on 1-manifolds, Proceedings Amer. Math. Soc. 122 (1994) 333–340
• R Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Math. 81, Birhauser (1984)
• R Zimmer, Actions of semisimple groups and discrete subgroups, Proc. Internat. Congr. Math. (Berkeley 1986), Vol 2 (A W Gleason, editor) Amer. Math. Soc. Providence, RI (1987) 1247–1258