The modular group action on real SL(2)–characters of a one-holed torus

For $t \in ℝ$ , the $Γ$ –action on $κ^{- 1} (t) \cap ℝ$ displays rich and varied dynamics. The action of $Γ$ preserves a Poisson structure defining a $Γ$ –invariant area form on each $κ^{- 1} (t) \cap ℝ$ . For $t < 2$ , the action of $Γ$ is properly discontinuous on the four contractible components of $κ^{- 1} (t) \cap ℝ$ 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 $\bar{M}$ . For $t = 2$ , the level set $κ^{- 1} (t) \cap ℝ$ consists of characters of reducible representations and comprises two ergodic components corresponding to actions of $GL (2, ℤ)$ on ${(ℝ ∕ ℤ)}^{2}$ and $ℝ^{2}$ respectively. For $2 < t \leq 18$ , the action of $Γ$ on $κ^{- 1} (t) \cap ℝ$ is ergodic. Corresponding to the Fricke space of a three-holed sphere is a $Γ$ –invariant open subset $Ω \subset ℝ^{3}$ whose components are permuted freely by a subgroup of index $6$ in $Γ$ . The level set $κ^{- 1} (t) \cap ℝ$ intersects $Ω$ if and only if $t > 18$ , in which case the $Γ$ –action on the complement $(κ^{- 1} (t) \cap ℝ) - Ω$ is ergodic.

Keywords

surface, fundamental group, character variety, representation variety, mapping class group, ergodic action, proper action, hyperbolic structure with cone singularity, Fricke space, Teichmüller space

Mathematical Subject Classification 2000

Primary: 57M05

Secondary: 20H10, 30F60

References

Forward citations

Publication

Received: 19 August 2001

Revised: 7 June 2003

Accepted: 10 July 2003

Published: 18 July 2003

Proposed: Walter Neumann

Seconded: Benson Farb, Martin Bridson

Authors

William M Goldman
Mathematics Department University of Maryland College Park Maryland 20742 USA

Volume 7, issue 1 (2003)

William M Goldman