Weighted cogrowth formula for free groups
Dynamical Systems
2018-02-26 v1 Probability
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.
Cite
@article{arxiv.1802.08361,
title = {Weighted cogrowth formula for free groups},
author = {Johannes Jaerisch and Katsuhiko Matsuzaki},
journal= {arXiv preprint arXiv:1802.08361},
year = {2018}
}