## Connective constants and height functions for Cayley graphs

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##### Publication Date

2017-03-31##### Journal Title

Transactions of the American Mathematical Society

##### ISSN

0002-9947

##### Publisher

American Mathematical Society

##### Volume

369

##### Issue

8

##### Pages

5961-5980

##### Language

English

##### Type

Article

##### This Version

AM

##### Metadata

Show full item record##### Citation

Grimmett, G., & Li, Z. (2017). Connective constants and height functions for Cayley graphs. Transactions of the American Mathematical Society, 369 (8), 5961-5980. https://doi.org/10.1090/tran/7166

##### Abstract

The connective constant $μ$($G$) of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called “unimodular graph height functions”. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a “$\textit{group}$ height function”. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency.
It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition $\textit{harmonic}$ on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic.
Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.

##### Keywords

self-avoiding walk, connective constant, vertex-transitive graph, quasi-transitive graph, bridge decomposition, Cayley graph, Higman group, graph height function, group height function, indicability, harmonic function, solvable group, unimodularity

##### Sponsorship

The first author was supported in part by EPSRC Grant EP/I03372X/1. The second author was supported in part by Simons Collaboration Grant #351813 and NSF grant #1608896.

##### Funder references

EPSRC (EP/I03372X/1)

##### Identifiers

External DOI: https://doi.org/10.1090/tran/7166

This record's URL: https://www.repository.cam.ac.uk/handle/1810/263104

##### Rights

Attribution-NonCommercial-NoDerivatives 4.0 International

Licence URL: http://creativecommons.org/licenses/by-nc-nd/4.0/