Path-connectivity of Thick Laminations, and Markov Processes with Thick Limit Sets

TL;DR

This paper proves path-connectivity properties of thick laminations on high-genus surfaces and explores implications for the Morse boundary of the mapping class group, including constructing a path-connected limit set of thick laminations.

Contribution

It establishes that any two thick laminations can be connected by a path of thick laminations and constructs a subshift with a path-connected limit set of thick laminations.

Findings

Any two epsilon-thick laminations can be joined by a delta-thick lamination path.

The Morse boundary of the mapping class group is path-connected.

A subshift of finite type with a path-connected limit set of thick laminations is constructed.

Abstract

A lamination $\lambda$ is $\epsilon$-thick (with respect to a basepoint $X$), if the Teichm\"uller ray from $X$ in the direction of $\lambda$ stays in the $\epsilon$-thick part. We show that, for surfaces of high enough genus, any two $\epsilon$-thick laminations can be joined by a path of $\delta$-thick laminations. As a consequence, we show that the Morse boundary of the mapping class group is path-connected. Furthermore, we construct a subshift of finite type on the mapping class group, whose limit set consists only of thick laminations and is path-connected.