Statistical Hyperbolicity in Teichmüller Space

Statistical Hyperbolicity in Teichmüller Space

In this paper we explore the idea that Teichmüller space with the Teichmüller metric is hyperbolic “on average.” We consider several different measures on Teichmüller space and show that with respect to each one, the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible.

Quantum Teichmüller Space

We describe explicitly a noncommutative deformation of the *-algebra of functions on the Teichmüller space of Riemann surfaces with holes equivariant w.r.t. the mapping class group action.

Statistical hyperbolicity in groups

In this paper, we introduce a geometric statistic called the sprawl of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (ie without this obstruction) are called statistically hyperbolic. We first relate sprawl to ...

Dynamics over Teichmüller Space

Fuzzy Subgroups and the Teichmüller Space

There exists a generalization of the Teichmüller space of a covering group. In this paper we combine this generalized Teichmüller space T (G) and any fuzzy subgroup A : G−→ F where G is a subgroup of the group consisting of such orientation preserving and orientation reversing Möbius transformations which act in the upper half-plane of the extended complex plane. A partially ordered set F = (F ...