We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a wordhyperbolic group into a semisimple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations , and discusses questions and open problems which arise.
Carathéodory's metrics on Teichmüller spaces and Lshaped pillowcases. Duke Math. J. 167 (2018), no. 3, 497–535.
One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each Teichmüller space of a closed surface of genus g≥2. The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential.
 Award ID(s):
 1800742
 Publication Date:
 NSFPAR ID:
 10104460
 Journal Name:
 Duke Math. J
 Volume:
 167
 Issue:
 3
 Page Range or eLocationID:
 497–535
 Sponsoring Org:
 National Science Foundation
More Like this


World Scientific (Ed.)In general, it is difficult to measure distances in the Weil–Petersson metric on Teichmüller space. Here we consider the distance between strata in the Weil–Petersson completion of Teichmüller space of a surface of finite type. Wolpert showed that for strata whose closures do not intersect, there is a definite separation independent of the topology of the surface. We prove that the optimal value for this minimal separation is a constant [Formula: see text] and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for [Formula: see text] and give a lower bound on the size of the gap between [Formula: see text] and the other distances. A major component of the paper is an effective version of Wolpert’s upper bound on [Formula: see text], the inner product of the Weil–Petersson gradient of length functions. We further bound the distance to the boundary of Teichmüller space of a hyperbolic surface in terms of the length of the systole of the surface. We also obtain new lower bounds on the systole for the Weil–Petersson metric on the moduli space of a punctured torus.

Abstract In this article, we propose an Outer space analog for the principal stratum of the unit tangent bundle to the Teichmüller space ${\mathcal{T}}(S)$ of a closed hyperbolic surface $S$. More specifically, we focus on properties of the geodesics in Teichmüller space determined by the principal stratum. We show that the analogous Outer space “principal” periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmüller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmüller space context.

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a oncepunctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of BenjaminiTessera and CantrellFurman show that scaling limits of such FPP are given by CarnotCarathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any CarnotCarathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugationinvariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugationinvariant condition is also a necessary condition in all cases where scaling limits are known to exist.