We study unimodular measures on the space
of all pointed
Riemannian
–manifolds.
Examples can be constructed from finite-volume manifolds, from
measured foliations with Riemannian leaves, and from invariant
random subgroups of Lie groups. Unimodularity is preserved under
weak limits,
and under certain geometric constraints (eg bounded geometry) unimodular measures can be
used to compactify sets of finite-volume manifolds. One can then understand the geometry of
manifolds
with large, finite volume by passing to unimodular limits.
We develop a structure theory for unimodular measures on
,
characterizing them via invariance under a certain geodesic flow, and showing that
they correspond to transverse measures on a foliated “desingularization” of
.
We also give a geometric proof of a compactness theorem for unimodular
measures on the space of pointed manifolds with pinched negative
curvature, and characterize unimodular measures supported on hyperbolic
–manifolds
with finitely generated fundamental group.
Keywords
unimodular measures, Benjamini–Schramm convergence,
invariant random subgroup, hyperbolic geometry, Lie group
