## 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
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### 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
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