Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces

TL;DR

This paper provides an elementary proof of the trace classification of conjugacy classes in SL(2,C), explores hyperbolic structures on simple surfaces, and computes their deformation spaces using trace coordinates.

Contribution

It offers a detailed proof of trace-based conjugacy classification and computes deformation spaces for specific hyperbolic surfaces, extending classical results.

Findings

Trace coordinates determine conjugacy classes in SL(2,C) for pairs and triples.

Deformation spaces of hyperbolic structures on certain surfaces are explicitly computed.

Connections between trace coordinates and hyperbolic geometry are clarified.

Abstract

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 representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated. We describe it in detail when n= 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.)