×

The modular group action on real \(SL(2)\)-characters of a one-holed torus. (English) Zbl 1037.57001

Let \(M\) be a compact oriented surface of genus one with one boundary component. The polynomial \[ \kappa(x,y,z)=x^2+y^2+z^2-xyz-2 \] arises in the definition of the locus of the space of representations of the fundamental group of \(M\) in \(SU(2,\mathbb C)\). Let \(\Gamma\) be the polynomial automorphism of \(\mathbb C^3\) that preserves the polynomial \(\kappa\). The author gives a complete classification of the dynamics of the action of \(\Gamma\) on the fibres \(\kappa^{-1}(t)\cap \mathbb R^3\) for all \(t\) in \(\mathbb R\). This action of \(\Gamma\) preserves a Poisson structure defining a \(\Gamma-\)invariant area form on each \(\kappa^{-1}(t)\cap \mathbb R^3\). The main result of the paper is the following
Theorem :
\(\bullet\) For \(t<-2\), the group \(\Gamma\) acts properly on \(\kappa^{-1}(t)\cap \mathbb R^3\);
\(\bullet\) For \(-2\leq t< 2\), there is a compact connected component \(C_t\) of \(\kappa^{-1}(t)\cap \mathbb R^3\) and \(\Gamma\) and \(\Gamma\) acts properly on the complement \(\kappa^{-1}(t)\cap \mathbb R^3-C_t\);
\(\bullet\) For \(t=2\), the action of \(\Gamma\) is ergodic on the compact subset \(\kappa^{-1}(2)\cap[-2,3]^3\) and the action is ergodic on the complement \(\kappa^{-1}(2)- [-2,2]^3\);
\(\bullet\) For \(2<t\leq 18\), the group \(\Gamma\) acts ergodically on \(\kappa^{-1}(t)\cap \mathbb R^3\);
\(\bullet\) for \(t>18\), the group \(\Gamma\) acts properly and freely on an open subset \(\Omega_t\subset \kappa^{-1}(t)\cap \mathbb R^3\), permuting its components. The \(\Gamma-\) action on the complement of \(\Omega_t\) is ergodic.
By a classical result of Horowitz, the action of the group \(\Gamma\) is commensurable with the action of the modular group of \(M\) (which is seen as the group of automorphisms of \(\pi_1(M)\)) on the space of equivalence classes of representations \(\pi_1(M)\to SL(2,\mathbb C)\).
The author gives an identification of the properly discontinuous action of \(\Gamma\) on \(\kappa^{-1}(t)\cap \mathbb R^3\), for various real numbers \(t\), with actions of the modular group on various Teichmüller spaces of (possibly singular) hyperbolic structures on the surface \(M\). For \(t>2\), the discussion involves the Fricke space of the three-holed sphere.
The paper is very interesting and well-written.

MSC:

57M05 Fundamental group, presentations, free differential calculus
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F60 Teichmüller theory for Riemann surfaces
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML EMIS

References:

[1] W Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics 820, Springer (1980) · Zbl 0452.32015
[2] L Bers, F P Gardiner, Fricke spaces, Adv. in Math. 62 (1986) 249 · Zbl 0634.32020
[3] B H Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. \((3)\) 77 (1998) 697 · Zbl 0928.11030
[4] G W Brumfiel, H M Hilden, \(\mathrm{SL}(2)\) representations of finitely presented groups, Contemporary Mathematics 187, American Mathematical Society (1995) · Zbl 0838.20006
[5] P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser (1992) · Zbl 0770.53001
[6] M Culler, P B Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. \((2)\) 117 (1983) 109 · Zbl 0529.57005
[7] R Feres, Dynamical systems and semisimple groups: an introduction, Cambridge Tracts in Mathematics 126, Cambridge University Press (1998) · Zbl 1057.22001
[8] R Fricke, Über die Theorie der automorphen Modulgrupper, Nachr. Akad. Wiss. Göttingen (1896) 91
[9] R Fricke, F Klein, Vorlesungen der Automorphen Funktionen I, Teubner (1897) · JFM 42.0452.01
[10] R Fricke, F Klein, Vorlesungen der Automorphen Funktionen II, Teubner (1912) · JFM 43.0529.08
[11] J Gilman, B Maskit, An algorithm for 2-generator Fuchsian groups, Michigan Math. J. 38 (1991) 13 · Zbl 0724.20033
[12] W Goldman, Discontinuous groups and the Euler class, PhD thesis, University of California, Berkeley (1980)
[13] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557 · Zbl 0655.57019
[14] W M Goldman, Ergodic theory on moduli spaces, Ann. of Math. \((2)\) 146 (1997) 475 · Zbl 0907.57009
[15] W Goldman, An exposition of results of Fricke, in preparation
[16] W M Goldman, W D Neumann, Homological action of the modular group on some cubic moduli spaces, Math. Res. Lett. 12 (2005) 575 · Zbl 1087.57001
[17] W J Harvey, Spaces of discrete groups, Academic Press (1977) 295
[18] R D Horowitz, Induced automorphisms on Fricke characters of free groups, Trans. Amer. Math. Soc. 208 (1975) 41 · Zbl 0306.20027
[19] G Kern-Isberner, G Rosenberger, Über Diskretheitsbedingungen und die Diophantische Gleichung \(ax^2+by^2+cz^2=dxyz\), Arch. Math. \((\)Basel\()\) 34 (1980) 481 · Zbl 0432.20044
[20] Y Imayoshi, M Taniguchi, An introduction to Teichmüller spaces, Springer (1992) · Zbl 0754.30001
[21] A Lubotzky, A R Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1985) · Zbl 0598.14042
[22] R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer (1977) · Zbl 0368.20023
[23] W Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980) 91 · Zbl 0433.20033
[24] W Magnus, A Karrass, D Solitar, Combinatorial group theory, Dover Publications (2004) · Zbl 1130.20307
[25] G A Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 17, Springer (1991) · Zbl 0732.22008
[26] R C McOwen, Prescribed curvature and singularities of conformal metrics on Riemann surfaces, J. Math. Anal. Appl. 177 (1993) 287 · Zbl 0806.53040
[27] J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures I, Ann. of Math. \((2)\) 120 (1984) 401 · Zbl 0583.57005
[28] C C Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966) 154 · Zbl 0148.37902
[29] S Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons (1988) · Zbl 0667.30040
[30] J Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1964) 385 · JFM 46.0175.01
[31] J Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927) 189 · JFM 53.0545.12
[32] G Stantchev, Dynamics of the modular group acting on \(GL(2,\mathbbR)\)-characters of a once-punctured torus, PhD thesis, University of Maryland (2003)
[33] J Stillwell, The Dehn-Nielsen theorem, Springer (1987)
[34] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793 · Zbl 0724.53023
[35] R J Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics 81, Birkhäuser Verlag (1984) · Zbl 0571.58015
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.