  We study the connective constants of weighted self-avoiding walks (SAWs) on
infinite graphs and groups. The main focus is upon weighted SAWs on finitely
generated, virtually indicable groups. Such groups possess so-called 'height
functions', and this permits the study of SAWs with the special property of
being bridges. The group structure is relevant in the interaction between the
height function and the weight function. The main difficulties arise when the
support of the weight function is unbounded, since the corresponding graph is
no longer locally finite.
  There are two principal results, of which the first is a condition under
which the weighted connective constant and the weighted bridge constant are
equal. When the weight function has unbounded support, we work with a
generalized notion of the 'length' of a walk, which is subject to a certain
condition.
  In the second main result, the above equality is used to prove a continuity
theorem for connective constants on the space of weight functions endowed with
a suitable distance function.
