Laminations with transverse measures in ordered abelian semigroups

TL;DR

This paper develops a framework for constructing and analyzing transverse measures on codimension-1 laminations using ordered algebraic structures, expanding the class of laminations that can be measured beyond traditional real-valued measures.

Contribution

It introduces new ordered semi-algebraic structures for defining measures on laminations and explores their applications, including the realization problem and actions on associated trees.

Findings

Laminations can admit transverse measures with values in ordered semirings.

Finite or infinite depth measured laminations are characterized by measures in a specific ordered semiring.

The realization problem links weight vectors on branched manifolds to laminations with transverse measures.

Abstract

We describe a construction of ordered algebraic structures (ordered abelian semigroups, ordered commutative semirings, etc.) and describe applications to codimension-1 laminations. For a suitable ordered semi- algebraic structure $\mathbb L$ and measurable space $X$ we define $\mathbb L$-measures $\nu$ on $X$. If $L$ is a codimension-1 lamination in a manifold, it often admits transverse $\mathbb L$-measures for some $\mathbb L$. Transverse $\mathbb L$-measures can be used to understand classes of laminations much larger than the class of laminations admitting transverse positive $\mathbb R$-measures. In particular, we show that "finite or infinite depth measured laminations" are laminations admitting transverse measures with values in a certain ordered semiring $\bar{\mathbb O}$ satisfying the additional property that locally the values lie in a smaller semiring $\mathbb P$. We consider the "realization problem:" In one version, this deals with the problem whether an $\mathbb P$-invariant weight vector assigned to a branched manifold $B$ (satisfying certain branch equations) determines a lamination $L$ carried by $B$ with a transverse $\bar{\mathbb O}$-measure inducing the weights on $B$. We describe further laminations which may not be $\mathbb L$-measured, but are "well-covered" by laminations with transverse $\mathbb L$-measures. We also investigate actions on $\mathbb L$-trees which are associated to essential laminations with transverse $\mathbb L$-measures. In appendices, we develop ideas about $\mathbb L$-measures a little further, for example showing that a $\mathbb P$-measure can be interpreted as a kind of probability measure.