  We study the cluster automorphism group of a skew-symmetric cluster algebra
with geometric coefficients. For this, we introduce the notion of gluing free
cluster algebra, and show that under a weak condition the cluster automorphism
group of a gluing free cluster algebra is a subgroup of the cluster
automorphism group of its principal part cluster algebra (i.e. the
corresponding cluster algebra without coefficients). We show that several
classes of cluster algebras with coefficients are gluing free, for example,
cluster algebras with principal coefficients, cluster algebras with universal
geometric coefficients, and cluster algebras from surfaces (except a 4-gon)
with coefficients from boundaries. Moreover, except four kinds of surfaces, the
cluster automorphism group of a cluster algebra from a surface with
coefficients from boundaries is isomorphic to the cluster automorphism group of
its principal part cluster algebra; for a cluster algebra with principal
coefficients, its cluster automorphism group is isomorphic to the automorphism
group of its initial quiver.
