  Infrared growth of geometrical fluctuations in inflationary spacetimes is
investigated. The problem of gauge-invariant characterization of growth of
perturbations, which is of interest also in other spacetimes such as black
holes, is addressed by studying evolution of the lengths of curves in the
geometry. These may either connect freely falling "satellites," or wrap
non-trivial cycles of geometries like the torus, and are also used in
diffeomorphism- invariant constructions of two-point functions of field
operators. For spacelike separations significantly exceeding the Hubble scale,
no spacetime geodesic connects two events, but one may find geodesics
constrained to lie within constant-time spatial slices. In inflationary
geometries, metric perturbations produce significant and growing corrections to
the lengths of such geodesics, as we show in both quantization on an inflating
torus and in standard slow-roll inflation. These become large, signaling
breakdown of a perturbative description of the geometry via such observables,
and consistent with perturbative instability of de Sitter space. In particular,
we show that the geodesic distance on constant time slices during inflation
becomes non-perturbative a few e-folds after a given scale has left the
horizon, by distances \sim 1/H^3 \sim RS, obstructing use of such geodesics in
constructing IR-safe observables based on the spatial geometry. We briefly
discuss other possible measures of such geometrical fluctuations.
