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2003 The modular group action on real $SL(2)$–characters of a one-holed torus
William M Goldman
Geom. Topol. 7(1): 443-486 (2003). DOI: 10.2140/gt.2003.7.443


The group Γ of automorphisms of the polynomial

κ ( x , y , z ) = x 2 + y 2 + z 2 x y z 2

is isomorphic to

PGL ( 2 , ) ( 2 2 ) .

For t, the Γ–action on κ1(t) displays rich and varied dynamics. The action of Γ preserves a Poisson structure defining a Γ–invariant area form on each κ1(t). For t<2, the action of Γ is properly discontinuous on the four contractible components of κ1(t) and ergodic on the compact component (which is empty if t<2). The contractible components correspond to Teichmüller spaces of (possibly singular) hyperbolic structures on a torus M¯. For t=2, the level set κ1(t) consists of characters of reducible representations and comprises two ergodic components corresponding to actions of GL(2,) on ()2 and 2 respectively. For 2<t18, the action of Γ on κ1(t) is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Γ–invariant open subset Ω3 whose components are permuted freely by a subgroup of index 6 in Γ. The level set κ1(t) intersects Ω if and only if t>18, in which case the Γ–action on the complement (κ1(t))Ω is ergodic.


Download Citation

William M Goldman. "The modular group action on real $SL(2)$–characters of a one-holed torus." Geom. Topol. 7 (1) 443 - 486, 2003.


Received: 19 August 2001; Revised: 7 June 2003; Accepted: 10 July 2003; Published: 2003
First available in Project Euclid: 21 December 2017

zbMATH: 1037.57001
MathSciNet: MR2026539
Digital Object Identifier: 10.2140/gt.2003.7.443

Primary: 57M05
Secondary: 20H10, 30F60

Rights: Copyright © 2003 Mathematical Sciences Publishers


Vol.7 • No. 1 • 2003
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