Weighted cogrowth formula for free groups
Johannes Jaerisch, and Katsuhiko Matsuzaki

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 endowed with variable edge lengths, by an arbitrary subgroup of . 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 to the Poincar\'e exponent of . Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Geometry and complex manifolds
