Laminations with transverse measures in ordered abelian semigroups

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.