The Modular Group Action on Real SL(2)-characters of a One-Holed Torus

By William M. Goldman. 2003

143 citations recorded by the source corpus.

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 18, in which case the Gamma-action on the complement ktR - Omega is ergodic.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/399df6492910c70d437bfbc7ccaa920604e89a1f

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Framed BPS states (2010, unknown): “For useful information on this moduli space see [60, 61].”
Generalized Markoff maps and McShane's identity (2005, background): “G T ] 2 2 Fe b for closed hyperbolic 3-manifolds which are obtained by doing hyperbolic Dehn surgery on such manifolds.”
Counting Mapping Class group orbits on hyperbolic surfaces (2016, background): “ng class group Mod1,1 acts on this set by Markoﬀ moves given by (∗∗∗). Note that for equation (3.3) parametrizes the character variety of one holed torus with a boundary component. By work of Goldman [G4] this action on each connected component of V (κ) deﬁned by (3.3) is either properly continuous ”
Markoff Triples and Strong Approximation (2015, background): “Theorem 1 in the weaker form that | C ( p ) | ∼ | X ∗ ( p ) | as p → ∞ , can be viewed as the ﬁnite ﬁeld analogue of [18], where it is shown that the action of (cid:2) on the compact real components of the character variety of the mapping class group of the once punctured torus is ergodic As in…”