Thurston geodesics: no backtracking and active intervals

TL;DR

This paper introduces the concept of active intervals for subsurfaces along Thurston geodesics, showing they behave like quasi-geodesics in the curve graph with bounded outside movement, extending previous Teichmuller space results.

Contribution

It defines active intervals in Thurston geometry and demonstrates their properties, including no backtracking and bounded outside movement, generalizing Teichmuller space concepts.

Findings

Active intervals are associated with subsurfaces along Thurston geodesics.

Short curves during active intervals form reparametrized quasi-geodesics in the curve graph.

Outside active intervals, the geodesic's movement in the curve graph is uniformly bounded.

Abstract

We develop the notion of the active interval for a subsurface along a geodesic in the Thurston metric on Teichmuller space of a surface S. That is, for any geodesic in the Thurston metric and any subsurface R of S, we find an interval of times where the length of the boundary of R is uniformly bounded and the restriction of the geodesic to the subsurface R resembles a geodesic in the Teichmuller space of R. In particular, the set of short curves in R during the active interval represents a reparametrized quasi-geodesic in the curve graph of R (no backtracking) and the amount of movement in the curve graph of R outside of the active interval is uniformly bounded which justifies the name active interval. These intervals provide an analogue of the active intervals introduced by the third author in the setting of Teichmuller space equipped with the Teichmuller metric.