  We generalise surface cluster algebras to the case of infinite surfaces where
the surface contains finitely many accumulation points of boundary marked
points. To connect different triangulations of an infinite surface, we consider
infinite mutation sequences.
  We show transitivity of infinite mutation sequences on triangulations of an
infinite surface and examine different types of mutation sequences. Moreover,
we use a hyperbolic structure on an infinite surface to extend the notion of
surface cluster algebras to infinite rank by giving cluster variables as lambda
lengths of arcs. Furthermore, we study the structural properties of infinite
rank surface cluster algebras in combinatorial terms, namely we extend "snake
graph combinatorics" to give an expansion formula for cluster variables. We
also show skein relations for infinite rank surface cluster algebras.
