Narrow Results

We introduce a family of multiplicative distributions {μ_s|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μ_s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

The notion of a generalized scale emerged in recent joint work with Afsar–Brownlowe–Larsen on equilibrium states on $C^{\ast }$-algebras of right Least Common Multiple (LCM) monoids, where it features as the key datum for the dynamics under investigation. This work provides the structure theory for such monoidal homomorphisms. We establish the uniqueness of the generalized scale and characterize its existence in terms of a simplicial graph arising from a new notion of irreducibility inside right LCM monoids. In addition, the method yields an explicit construction of the generalized scale if existent. We discuss applications for graph products as well as algebraic dynamical systems and reveal a striking connection to Saito’s degree map.

Suppose $X$ and $Y$ are finite complexes, with $Y$ simply connected. Gromov conjectured that the number of mapping classes in $[X,Y]$ which can be realized by $L$-Lipschitz maps grows asymptotically as $L^{\alpha }$, where $\alpha$ is an integer determined by the rational homotopy type of $Y$ and the rational cohomology of $X$. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is $L^8$ but the true growth is $L^{8}logL$. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number $r\ge 4$, there is a pair $X,Y$ for which the growth of $[X,Y]$ is essentially $L^r$.