×

zbMATH — the first resource for mathematics

Distortion elements in group actions on surfaces. (English) Zbl 1088.37009
Here, \(S\) is a closed orientable surface, \(\mu\) is a Borel probability measure on \(S\) and \(\text{Diff}(S)_0\) is the group if \(C^1\)-diffeomorphisms of \(S\) that are isotopic to the identity. Let \(G\) be a finitely generated group. An element \(f\in G\) is said to be a distorsion element of \(G\) if \(f\) has infinite order and if we have \(\lim\inf_{n\to\infty} | f^n| / n=0\) where \(| f^n| \) denotes the length of the element \(f^n\) of \(G\) with respect to the word metric associated to some finite system of generators of \(G\). (The definition obviously does not depend on the chosen system of generators). The group \(G\) is said to satisfy property \(T\) if the identity element of \(G\) is isolated in the unitary dual \(\widehat G\). The group \(G\) is said to be almost simple if every normal subgroup of \(G\) is finite or has finite index. The main results of this paper are the following theorems:
Theorem 1: If \(f\) is a distorsion element of \(\text{Diff}(S)_0\) and if \(\mu\) is \(f\)-invariant, then the support of \(\mu\) is contained in the fixed-point set of \(f\), provided the genus of \(S\) is \(\geq 2\).
The authors also give related results in the case where \(S\) is a torus or a 2-sphere.
Theorem 2: If \(S\) has positive genus, if \(G\) is almost simple and possesses a distorsion element \(u\), and if either the Borel measure \(\mu\) has infinite support or \(G\) satisfies property \(T\), then any homomorphism from \(G\) into the group of \(C^1\)-diffeomorphisms of \(S\) that preserve \(\mu\) has finite image.
Again, the authors give a related result in the case where \(S\) is the 2-sphere.
Theorem 3: If \(G\) is almost simple and has a subgroup isomorphic to the three-dimensional Heisenberg group, then any homomorphism from \(G\) into the group of \(C^1\)-diffeomorphisms of \(S\) that preserve \(\mu\) has finite image.
Several corollaries are obtained, in particular concerning the images of homomorphisms from subgroups of higher-rank lattices in the group of \(C^1\)-diffeomorphisms of \(S\) that preserve \(\mu\).

MSC:
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57M60 Group actions on manifolds and cell complexes in low dimensions
22F10 Measurable group actions
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] E. Alibegović, Translation lengths in \(\mathrm Out(F_n)\), Geom. Dedicata 92 (2002), 87–93. · Zbl 1041.20024 · doi:10.1023/A:1019695003668
[2] M. Brown and J. M. Kister, Invariance of complementary domains of a fixed point set, Proc. Amer. Math. Soc. 91 (1984), 503–504. JSTOR: · Zbl 0547.57010 · doi:10.2307/2045329 · links.jstor.org
[3] M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS) 1 (1999) 199–235. · Zbl 0932.22008 · doi:10.1007/s100970050007
[4] D. Calegari, personal communication, 2004.
[5] B. Farb, A. Lubotzky, and Y. Minsky, Rank-\(1\) phenomena for mapping class groups, Duke Math. J. 106 (2001), 581–597. · Zbl 1025.20023 · doi:10.1215/S0012-7094-01-10636-4
[6] B. Farb and H. Masur, Superrigidity and mapping class groups, Topology 37 (1998), 1169–1176. · Zbl 0946.57018 · doi:10.1016/S0040-9383(97)00099-2
[7] B. Farb and P. Shalen, Real-analytic actions of lattices, Invent. Math. 135 (1999), 273–296. · Zbl 0954.22007 · doi:10.1007/s002220050286
[8] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8 *, Charles Conley memorial issue (1988), 99–107. · Zbl 0634.58023 · doi:10.1017/S0143385700009366
[9] -, “Rotation numbers for area preserving homeomorphisms of the open annulus” in Dynamical Systems and Related Topics (Nagoya, Japan, 1990) , Adv. Ser. Dynam. Systems 9 , World Sci., River Edge, N. J., 1991, 123–127.
[10] -, Rotation vectors and fixed points of area preserving surface diffeomorphisms, Trans. Amer. Math. Soc. 348 (1996), 2637–2662. JSTOR: · Zbl 0862.58006 · doi:10.1090/S0002-9947-96-01502-4 · links.jstor.org
[11] J. Franks and M. Handel, Area preserving group actions on surfaces, Geom. Topol. 7 (2003), 757–771. · Zbl 1036.37010 · doi:10.2140/gt.2003.7.757 · emis:journals/UW/gt/GTVol7/paper21.abs.html · eudml:123487
[12] -, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol. 7 (2003), 713–756. · Zbl 1034.37028 · doi:10.2140/gt.2003.7.713 · emis:journals/UW/gt/GTVol7/paper20.abs.html · eudml:123400
[13] S. M. Gersten and H. B. Short, Rational subgroups of, biautomatic groups, Ann. of Math. (2) 134 (1991), 125–158. JSTOR: · Zbl 0744.20035 · doi:10.2307/2944334 · links.jstor.org
[14] É. Ghys, Sur les groupes engendrés par des difféomorphismes proches de l’\(\!\)identité, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), 137–178. · Zbl 0809.58004 · doi:10.1007/BF01237675
[15] M. Handel, A fixed-point theorem for planar homeomorphisms, Topology 38 (1999), 235–264. · Zbl 0928.55001 · doi:10.1016/S0040-9383(98)00001-9
[16] D. A. Kazhdan [D. A. KažDan], On the connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967), 63–65.; translation from Funkcional. Anal. i Priložen. 1 (1967), 71–74. · Zbl 0168.27602 · doi:10.1007/BF01075866
[17] A. Lubotzky, S. Mozes, and M. S. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. 91 (2000), 5–53. · Zbl 0988.22007 · doi:10.1007/BF02698740 · numdam:PMIHES_2000__91__5_0 · eudml:104167
[18] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. (3) 17 , Springer, Berlin, 1991. · Zbl 0732.22008
[19] G. Mess, personal communication, 2004.
[20] L. Polterovich, Growth of maps, distortion in groups and symplectic geometry, Invent. Math. 150 (2002), 655–686. · Zbl 1036.53064 · doi:10.1007/s00222-002-0251-x
[21] W. P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347–352. · Zbl 0305.57025 · doi:10.1016/0040-9383(74)90025-1
[22] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monogr. Math. 81 , Birkhäuser, Basel, 1984. · Zbl 0571.58015
[23] -, “Actions of semisimple groups and discrete subgroups” in Proceedings of the International Congress of Mathematicians, Vol. 2 (Berkeley, 1986), Amer. Math. Soc., Providence, 1987, 1247–1258. · Zbl 0671.57028
[24] -, “Lattices in semisimple groups and invariant geometric structures on compact manifolds” in Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984) , Progr. Math. 67 , Birkhäuser, Boston, 1987, 152–210. · Zbl 0663.22008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.