# Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces

@article{Goldman2009TraceCO, title={Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces}, author={William M. Goldman}, journal={arXiv: Geometric Topology}, year={2009}, pages={611-684} }

The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a…

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## References

SHOWING 1-10 OF 85 REFERENCES

### Algebraic representations of Teichmüller space

- Mathematics
- 1994

This is a summary of the paper [S5] with the same title. It is divided in two parts. In part I, we summarize some general results on the character variety of representations of a finitely generated…

### Arithmetic of hyperbolic 3-manifolds

- Mathematics
- 2002

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an real…

### A rough fundamental domain for Teichmüller spaces

- Mathematics
- 1977

Let T(S) be the Teichmuller space of Riemann surfaces of finite type and let M(S) be the corresponding modular group. In [11] we described T(S) in terms of real analytic parameters. In this paper we…

### The Fenchel-Nielsen deformation

- Mathematics
- 1982

The uniformization theorem provides that a Riemann surface S of negative Euler characteristic has a metric of constant curvature -1. A hyperbolic structure can be understood in terms of its…

### On the symplectic geometry of deformations of a hyperbolic surface

- Mathematics
- 1983

Let R be a Riemann surface. In this manuscript we consider a geometry on the moduli space X(R) for R, which we regard as the space of equivalence classes of constant curvature metrics on the…

### Homological action of the modular group on some cubic moduli spaces

- Mathematics
- 2004

We describe the action of the automorphism group of the complex cubic x + y + x − xyz − 2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured…

### Invariants of 2 by 2 matrices, irreducible SL(2,C) characters and the Magnus trace map

- Mathematics
- 2006

We obtain an explicit characterization of the stable points of the action of G=SL(2,C) on the cartesian product G^n by simultaneous conjugation on each factor, in terms of the corresponding invariant…

### Characters of Free Groups Represented in the Two-Dimensional Special Linear Group*

- Mathematics
- 1972

We consider here the problem of determining when two elements in a free group will have the same character under all possible representations of the given group in the special linear group of 2 x 2…

### Topological components of spaces of representations

- Mathematics
- 1988

Since rt is a finitely generated group, the space Hom0r, G) is a real analytic variety whenever G is a connected Lie group, and is a real affine algebraic variety whenever G is a linear algebraic…