Trace polynomial for homotopy classes of simple closed curves on the twice punctured torus

1. Trace polynomial for homotopy classes of simple closed curves on the twice punctured torus Raquel Águeda Maté UCLM (Spain)

2. L. Bers (60’s): Transition from discrete to non-discrete groups? D. Mumford and D. Wright (80’s): First computer explorations. L. Keen and C. Series (late 80’s) Introduction

3. D. Mumford, C. Series and D. Wright, Indra’s Pearls, CUP (2002)

4. Deformation space of free Kleinian groups Γ. Γ Kleinian group: discrete subgroup of PSL(2, ). Γ generated by two elements: parabolic and loxodromic (trace ±2 and complex resp.). Introduction

5. Discrete, faithful and type-preserving representations of the fundamental group of a hyperbolic 3-manifold M in PSL(2, ) : ρ: π1(M) PSL(2, ) M is obtained by pinching a non-dividing curve in a handlebody of genus 2. Introduction

6. 3 Introduction

7. Γ acts properly discontinuously on (Γ discrete group). The action of Γ divides on: 1. Λ(Γ) = limit set, 2. Ω(Γ) = \ Λ(Γ) = regular set. Γ acts properly discontinuously on Ω(Γ). Mp ( 3 ()) /  H Introduction

8. The convex hull (Γ) of the limit set is the closure of the set geodesics that join each pair of points of Λ(Γ) in . (Γ) / Γ istheconvex core: the smallest convex set that contains all the geodesics in / Γ. The boundary of (Γ) is a pleated surface: pieces of hyperbolic planes that meet along a geodesic lamination. Pleating locus invariant by Γ. Invariant of the metric in .  ( (Γ)/ Γ) is homeomorphic to M . Introduction

9. Introduction (Γ) C Ax(g) 

10. Introduction Y. Y. Minsky Minsky (en (en Indra’s Pearls Indra’s Pearls ) )

11. Introduction Y. Y. Minsky Minsky (en (en Indra’s Pearls Indra’s Pearls ) )

12. Pleating variety, : locus in where is pleated along . is the locus where the curves in have real trace (Y. Choi and C. Series). (C()/Γ) can be calculated by studying pleating varieties. : some curve in becomes parabolic. Introduction (X,Y )  :  ((X,Y) ) / (X,Y) 2 C = R is a twice punctured torus

13. 4 4 Introduction

14. 1 Introduction

15. Analysis Discrete and faithful representations ρ: π1(M) PSL(2, ) Geometry Combinatorics Curve systems in M Introduction o Geometric structures in M Measured laminations: bending locus of the convex core boundary of M.

16. Aim Find the parameter space and its boundary . R / R Idea Introduction Use pleating variaties.

17. Homotopy classes of curve systems

18. Definition A curve system in Mis a family of disjoint essential simple loops whose components are homotopically distinct. Homotopy classes of curve systems

19. Parametrization of ∂M-homotopy classes of curve systems [Keen - Parker - Series] 2 1 1 Homotopy classes of curve systems

20. Normalize the set of generators of Γρ. Homotopy classes of curve systems Set up a set of generators for Γρ. B =  oi* (1), A =  oi* (1), Id =  oi*(2)

21. B A-1 A 2 2 1 3 2 2 C-1 C 1 1 2 2 1 B-1 1 1 2 2 Homotopy classes of curve systems Example ρ ( γ1 ) = BA-1BC-1AB-1B-1C ρ ( γ2 ) = BC-1

22. yi xi yi xi ui ui yi yi xi xi Homotopy classes of curve systems

23. 1 2 2 3 2 2 2 1 Homotopy classes of curve systems p ( ) = ( 5, 3, 2, 1 )

24. We find in every M-homotopy class a representative curve system in M, unique up to ∂M-homotopy, such that q1 ( ) ≤ q2 ( ) and 0 ≤ p2 ( ) ≤ q2 ( ). Homotopy classes of curve systems When are homotopic in M two curve systems which are non-homotopic in ∂M ?

25. : homeomorphisms of that extend to M and are homotopic to the identity. 1. Extensions of Dehn twists along compressible curves. 2. Hyperelliptic involution. Homotopy classes of curve systems

26. 1. Extensions of Dehn twists around compressible curves: α2, δ and ταn (δ), n Homotopy classes of curve systems 2  1

27. 1 2 Homotopy classes of curve systems 2. Hyperelliptic involution f. 2 1

28. THEOREM Homotopy classes of curve systems

29. Proposition Let a curve system in M and g a homeomorphism of M, then q2 ( g ( )) = q2 ( ). Reducing process: q1 ( g ( )) < q1 ( ), while q2 ( g ( )) = q2 ( ). Homotopy classes of curve systems

30. Proposition If q2 < q1, p1 > 0 and p1 < q1, then: Homotopy classes of curve systems

31. Homotopy classes of curve systems Proof

32. Homotopy classes of curve systems Corollary

33. ¿What if q1 does not get reduced by using τδ? Homotopy classes of curve systems Proposition

34. æ g is: Homotopy classes of curve systems Proposition

35. Homotopy classes of curve systems

36. Homotopy classes of curve systems Proposition

37. Trace polynomial

38. We compute the first two terms of the trace polynomial of curves in the canonic form in terms of their p-coordinates. Trace polynomial Which is the trace polynomial?

39. THEOREM Let γ be a simple closed curve on M with p-coordinates p(γ) = ( q1, q2, p1, p2 )where q1 ≤ q2 ≠ 0 and 0 ≤ p2 < q2. Let ρ(γ) the image of the curve γ by the representation ρ in PSL(2, ). Then the top term of the trace polynomial of ρ(γ) is a homogeneous polynomial of degree q2 on the complex variables X and Y with the following form: Furthermore, the term of degree q2 1 vanishes. Trace polynomial

40. Proof Trace polynomial By induction on

41. In the conditions of last theorem: Trace polynomial Corollary

42. Trace polynomial for homotopy classes of simple closed curves on the twice punctured torus Raquel Águeda Maté UCLM (Spain)