Secondary LS category of measured laminations

TL;DR

This paper introduces the secondary measured category for measured laminations, a new invariant that captures subtle properties of foliated spaces when the primary measured category vanishes, linking it to group growth.

Contribution

It defines the secondary measured category, proves classical results for it, and relates it to the growth of (pseudo)groups, especially in free suspensions by Rohlin groups.

Findings

Secondary measured category is a transverse invariant.

It vanishes when the primary measured category is zero.

In free suspensions, it equals the growth of the associated group.

Abstract

In the author's Ph.D., a version of the tangential LS category for foliated spaces depending on a transverse invariant measure, called the measured category, was introduced. Unfortunately, the measured category vanishes easily. When it is zero, the rate of convergence to zero of the quantity involved in the definition, by taking arbitrarily large homotopies, gives a new invariant, called the secondary measured category. Several versions of classical results are proved for the secondary measured category. It is also shown that the secondary measured category is a transverse invariant related to the growth of (pseudo)groups. The equality between secondary category and the growth of a group is done in the case of free suspensions by Rohlin groups.