Prescribing the behavior of Weil–Petersson geodesics in the moduli space of Riemann surfaces

Abstract

We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of geodesics is rich enough to provide for examples of closed WP geodesics in the thin part of the moduli space, as well as divergent WP geodesic rays with minimal filling ending lamination.Some ingredients of independent interest are the following: A strength version of Wolpert's Geodesic Limit Theorem proved in §4. The stability of hierarchy resolution paths between narrow pairs of partial markings or laminations in the pants graph proved in §5. A kind of symbolic coding for laminations in terms of subsurface coefficients presented in §7.and for every annular subsurface A(γ) with core curve γ inside Z,This condition is a local version of the bounded combinatorics of the end invariant in [BMM11]. A WP geodesic with bounded combinatorics end invariant stays in the thick part of the moduli space, [BMM11]. In the direction of Conjecture 1.3 we prove: Theorem 6.8. (Short Curve) Given A, R, R > 0 and a sufficiently small > 0, there is a constantw =w(A, R, R , ) with the following property. Let g : [a, b] → Teich(S) be a WP geodesic segment with A−narrow end invariant (ν − , ν + ). Let ρ : [m, n] → P (S) be a hierarchy path between ν − and ν + . Suppose that Z a large component domain of ρ has (R, R )−bounded combinatorics over an interval [m , n ] ⊂ J Z .If n − m ≥ 2w, then for every α ∈ ∂Z we have α (g(t)) ≤ for every t ∈ [a , b ], where a and b are the corresponding times to m +w and n −w, respectively. Proof of part (1):We show that after possibly passing to a subsequence there is an 1 ≤ i ≤ k and a curve γ ∈ σ i −τ , such that α n = ϕ −1 i,n (γ). Part (1) then follows from (4.9).Let J n ⊂ [s, T − s] be the subintervals in the statement of part (1). After possibly passing to a subsequence we may assume that the intervals J n converge to an interval J. Since each J n ⊂ [s, T − s] we have that J ⊂ [s, T − s]. By Theorem 4.5(3), for each j = 0, ..., k + 1, ϕ j,n (ζ n | [t j ,t j+1 ] ) →ζ| [t j ,t j+1 ] as n → ∞. Moreover ϕ j,n is an isometry of the WP metric. So the length of geodesics segments ζ n (J n ) converge to the length ofζ(J). Then since ζ n

Asymptotics of a class of Weil–Petersson geodesics and divergence of Weil–Petersson geodesics

“…each connected component of S\Z is an annulus or a three holed sphere. In [Mod15] we proved that any hierarchy path between a narrow pair is stable in the pants graph.…”

“…For the proof of Theorem 1.1 we use the control of the length-functions along WP geodesics developed in [Mod15] and ruled surfaces as in [BMM10]. The new ingredient here is the strict uniform contraction property of the nearest point projection to WP geodesic segments close to the thick part of a stratum which is not the product of lower complexity strata; see § 5, in particular, Theorems 5.1 and 5.14.…”

“…The following strengthened version of Wolpert's Geodesic Limit Theorem (see [Wol03] and [BMM11]) proved in §4 of [Mod15] provides a limiting picture for a sequence of bounded length WP geodesic segments in the Teichmüller space. We need this result for compactness arguments in §5.…”

Limit Sets of Weil–Petersson Geodesics

“…Note that pr j (X 1 ) and pr j (X n ) are in Teich(S j ). Claim 4.9 in the proof of Theorem 4.6 in [Mod15] tells us that for Bers markings µ(pr j (X 1 )) and µ(pr j (X n )) and curves β n = (ϕ j 1,n ) −1 (β),…”

Recurrent Weil–Petersson geodesic rays with non-uniquely ergodic ending laminations

“…Hierarchy paths have properties encoded in their end points and the associated subsurface coefficients. For a list of these properties see [BMM11,§2] and [Mod,§2]. Here we only state a key feature of hierarchy paths which is the no backtracking property.…”