单孔环面实SL（2）-特征标上的模群作用
The Modular Group Action on Real SL(2)-characters of a One-Holed Torus
摘要：
多项式kappa（x，y，z）= x^2 + y^2 + z^2 - xyz-2的自同构群Gamma同构于PGL（2，Z）与（Z/2+Z/2）的半直积.对于R中的t，ktR = kappa^{-1}（t）intersect R上的Gamma作用表现出丰富多样的动力学。Gamma的作用保留了Poisson结构，该结构定义了每个ktR上的Gamma不变面积形式。当t < 2时，Gamma的作用在ktR的四个可压缩分量上是适当不连续的，并且在紧分量上是遍历的（当t < -2时，紧分量是空的）。可收缩分量对应于环面M杆上双曲结构的Teichmueller空间（可能是奇异的）。当t = 2时，水平集ktR由可约表示的特征标组成，包含两个遍历分量，分别对应GL（2，Z）在（R/Z）^2和R^2上的作用.当2 < t <= 18时，Gamma对ktR的作用是遍历的.对应于一个三孔球的普朗克空间的是一个伽玛不变开子集Ω子集R^3，它的分量可以被伽玛中一个指数为6的子群自由置换。水平集ktR与Omega相交当且仅当t > 18，在这种情况下，对补ktR - Omega的Gamma作用是遍历的。
英文摘要：
The group Gamma of automorphisms of the polynomial kappa(x,y,z) = x^2 + y^2 + z^2 - xyz -2 is isomorphic to PGL(2,Z) semi-direct product with (Z/2+Z/2). For t in R, Gamma-action on ktR = kappa^{-1}(t) intersect R displays rich and varied dynamics. The action of Gamma preserves a Poisson structure defining a Gamma-invariant area form on each ktR. For t < 2, the action of Gamma is properly discontinuous on the four contractible components of ktR and ergodic on the compact component (which is empty if t < -2). The contractible components correspond to Teichmueller spaces of (possibly singular) hyperbolic structures on a torus M-bar. For t = 2, the level set ktR 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 <= 18, the action of Gamma on ktR is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Gamma-invariant open subset Omega subset R^3 whose components are permuted freely by a subgroup of index 6 in Gamma. The level set ktR intersects Omega if and only if t > 18, in which case the Gamma-action on the complement ktR - Omega is ergodic.
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