[Paper Review] On type-preserving representations of the thrice punctured projective plane group

Sara Maloni, Frédéric Palesi+1 more|arXiv (Cornell University)|Jul 22, 2018

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper studies type-preserving representations of the fundamental group of the thrice-punctured projective plane into PGL(2,R), using extended Kashaev's coordinates on decorated character varieties and balanced triangulations. It proves that in Euler class ±1, there are 6 connected components where all 2-sided simple closed curves map to hyperbolic elements, and 2 components where some are non-hyperbolic, resolving a question by Bowditch. The mapping class group acts ergodically on most components, with explicit trace formulas and geometric dynamics on ellipses in coordinate space.

ABSTRACT

In this paper we consider type-preserving representations of the fundamental group of the three--holed projective plane into $\mathrm{PGL}(2, \R) =\mathrm{Isom}(\HH^2)$ and study the connected components with non-maximal euler class. We show that in euler class zero for all such representations there is a one simple closed curve which is non-hyperbolic, while in euler class $\pm 1$ we show that there are $6$ components where all the simple closed curves are sent to hyperbolic elements and $2$ components where there are simple closed curves sent to non-hyperbolic elements. This answer a question asked by Brian Bowditch. In addition, we show also that in most of these components the action of the mapping class group on these non-maximal component is ergodic. In this work, we use an extension of Kashaev's theory of decorated character varieties to the context of non-orientable surfaces.

Motivation & Objective

• To determine the number of connected components of the type-preserving PGL(2,R)-character variety for the thrice-punctured projective plane, particularly in non-maximal Euler classes.
• To resolve Bowditch's question on whether representations exist where all 2-sided simple closed curves are hyperbolic but the representation is not discrete and faithful.
• To investigate the ergodicity of the mapping class group action on non-maximal Euler class components of the character variety.
• To extend Kashaev’s decorated character variety framework to non-orientable surfaces, enabling trace computations and geometric analysis.

Proposed method

• The authors use an extension of Kashaev’s theory of decorated character varieties to non-orientable surfaces, introducing triangle and trace coordinates for representations into PGL(2,R).
• They employ balanced triangulations of the surface N_{1,3} to define a coordinate system that respects the topology and holonomy of the punctures.
• Key equations relate the traces of generators to the Euler class, including the identity: (a+b−(c+d))² = (ab+4)(cd+4), and similar forms for other pairings.
• The trace of a closed curve is computed via trace formulas derived from the coordinate system, allowing classification of hyperbolicity or ellipticity of images of simple closed curves.
• The dynamics of Dehn twists are analyzed via their action on 2D ellipses in R⁴, showing that compositions of Dehn twists act as rotations with ergodic action for almost all parameter values.
• Ergodicity of the mapping class group is established by showing that gradients of invariant ellipses span the tangent space at generic points in the component.

Experimental results

Research questions

• RQ1 How many connected components exist in the PGL(2,R)-character variety of the thrice-punctured projective plane for Euler class e(ρ) = ±1 and e(ρ) = 0?
• RQ2 Are there type-preserving representations where all 2-sided simple closed curves are mapped to hyperbolic elements, even if the representation is not discrete and faithful, and if so, in which components do they occur?
• RQ3 Does the mapping class group act ergodically on the connected components of the character variety for non-maximal Euler classes?
• RQ4 Can Kashaev’s decorated character variety framework be extended to non-orientable surfaces with consistent trace formulas and geometric interpretation?
• RQ5 What is the dynamical behavior of Dehn twists on the character variety, particularly in relation to ergodicity and invariant sets?

Key findings

• In Euler class e(ρ) = 0, every type-preserving representation has at least one 2-sided simple closed curve mapped to a non-hyperbolic element.
• For Euler class e(ρ) = ±1, there are exactly 6 connected components where all 2-sided simple closed curves are sent to hyperbolic elements.
• In Euler class e(ρ) = ±1, there are exactly 2 connected components where some 2-sided simple closed curves are sent to non-hyperbolic elements.
• The mapping class group acts ergodically on the 6 components where all 2-sided curves are hyperbolic for e(ρ) = ±1.
• The action is also shown to be ergodic on the e(ρ) = 0 component, based on transverse dynamics on invariant ellipses in R⁴.
• The Dehn twist along a curve acts as a rotation on a 2D ellipse in the coordinate space, and the composition of two such twists generates a dense orbit for almost all parameter values.