Weighted cogrowth formula for free groups

TL;DR

This paper generalizes Grigorchuk's cogrowth formula to free groups with variable edge lengths, linking spectral properties of weighted Laplacians to subgroup growth rates using Patterson-Sullivan theory.

Contribution

It introduces a weighted cogrowth formula for free groups with variable edge lengths, connecting spectral and geometric indices via Patterson-Sullivan theory.

Findings

Generalized cogrowth formula for variable edge lengths

Established relationship between Laplacian spectrum and Poincaré exponent

Extended Patterson-Sullivan theory to weighted Cayley graphs

Abstract

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the Poincar\'e exponent of $G$. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.