Limit elements in the configuration algebra for a cancellative monoid

TL;DR

This paper introduces new spaces related to the growth functions of cancellative monoids and explores their structure through a fibration, revealing connections between pre-partition functions, opposite series, and residues of growth functions.

Contribution

It defines the spaces $al(Gamma,G)$ and $al(P_{Gamma,G})$, and establishes a fibration linking these spaces, providing a new framework to analyze growth functions of cancellative monoids.

Findings

The fibration $al_Omega$ is equivariant with a $$-action.

Sum of pre-partition functions in a fiber relates to residues of growth functions.

Under mild assumptions, the structure of these spaces is well-understood and transitive.

Abstract

We introduce two spaces $\Omega(\Gamma,G)$ and $\Omega(P_{\Gamma,G})$ of pre-partition functions and of opposite series, respectively, which are associated with a Cayley graph $(\Gamma,G)$ of a cancellative monoid $\Gamma$ with a finite generating system $G$ and with its growth function $P_{\Gamma,G}(t)$. Under mild assumptions on $(\Gamma,G)$, we introduce a fibration $\pi_\Omega:\Omega(\Gamma,G)\to \Omega(P_{\Gamma,G})$ equivariant with a $\Z_{\ge0}$-action, which is transitive if it is of finite order. Then, the sum of pre-partition functions in a fiber is a linear combination of residues of the proportion of two growth functions $P_{\Gamma,G}(t)$ and $P_{\Gamma,G}\mathcal{M}(t)$ attached to $(\Gamma,G)$ at the places of poles on the circle of the convergent radius.