Dynamics of the Automorphism Group of the GL(2,R)-Characters of a Once-puncutred Torus

Details Dynamics of the Automorphism Group of the GL(2,R)-Characters of a Once-puncutred Torus Dynamics of the Automorphism Group of the GL(2,R)-Characters of a Once-puncutred Torus

2003-09-04

arxiv.org/abs/math/0309072v1

Mathematics

Differential Geometry

52 pages, 14 figures; submitted to the Journal of Experimental Mathematics

Scientific paper

Let pi be a free group of rank 2. Its outer automorphism group Out(pi) acts on the space of equivalence classes of representations in Hom(pi, SL(2,C)). Let SLm(2,R) denote ths subset of GL(2,R) consisting of matrices of determinant -1 and let ISL(2,R) denote the subgroup (SL(2,R) union i SLm(2,R)) of SL(2,C). The representation space Hom(pi, ISL(2,R)) has four connected components, three of which consist of representations that send at least on generator of pi to iSLm(2,R). We investigate the dynamics of the Out(pi)-action on these components. The group Out(pi) is commensurable with the group Gamma of automorphisms of the polynomial kappa(x,y,z) = -x^2 - y^2 + z^2 + xyz -2. We show that for -14 < c < 2, the action of Gamma is ergodic on the level sets kappa^(-1)(c). For c < -14 the group Gamma acts properly and freely on an open subset OmegaMc of kappa^(-1)(c) and acts ergodically on the complement of OmegaMc. We construct an algorithm which determines, in polynomial time, if a point (x,y,z) in R^3 is Gamma-equivalent to a point in OmegaMc or in its complement. Conjugacy classes of ISL(2,R)-representations identify with R^3 via an appropriate restriction of the Fricke character map. Corresponding to the Fricke spaces of the once-punctures Klein bottle and the once-punctured Moebius band are Gamma-invariant open subsets OmegaK and OmegaM respectively. We give an explicit parametrization of OmegaK and OmegaM as subsets of R^3 and we show that OmegaM has a non-empty intersection with kappa^(-1)(c) if and only if c<-14, while OmegaK has a non-empty intersection with kappa^(-1)(c) if and only if c>6.

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