Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weighted cogrowth formula for free groups   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.

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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th