  The connective constant $\mu(G)$ 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 '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 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.
