  Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply
connected complete Riemannian manifold of non-positive sectional curvature or a
locally finite tree). Isometric actions of G on M are (by definition) points in
the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems:
1. The cocompact actions form an open subset of R. 2. The cocompact actions
with discrete orbits whose point-stabilizers have type F_n form an open subset
of the subspace of R consisting of all actions with discrete orbits. (F_1 means
finitely generated, F_2 means finitely presented etc.) The key idea is to
introduce a new "controlled topology" invariant of such actions - dependent on
n - which is unfamiliar when the orbits are not discrete but which becomes
familiar (cf 2.) when the orbits are discrete. (This is the first of two
papers.)

Group Theory;Geometric Topology