Affine and hyperbolic laminations
1.1. Affine plane and hyperbolic space
1.1.1. Pointed at infinity hyperbolic spaces. Let H
be the 3-dimensional
hyperbolic space. The sphere at infinity 9H
is the boundary of the visibility com-
pactification of H
: an escaping to infinity sequence of points hn G H
in this compactification iff the directing vectors of the geodesic rays [o, hn] issued
from a certain (= any) reference point o G H 3 converge.
This and all other facts formulated in this section can be easily checked in terms
of the upper half-space model of H 3 described in §1.1.2.
A hyperbolic space H 3 with a distinguished boundary point
q G 9H3 is called pointed at infinity. A
([Ka90]). Let (H3,?) be a pointed at infinity hyperbolic
space. The choice of the point q G 9H 3 determines the Busemann cocycle f3q on
H 3 x H 3 by the formula
(1.3) Pq(huh2) = lim[dist(/ii,ft) -dist(h2,h)] ,
where h G H 3 converges to q in the visibility topology. A
In other words, f3q(h\, h2) can be considered as a "regularization" of the formal
expression dist(hi,q) dist(h2,q). The Busemann cocycle obviously satisfies the
cocycle chain rule:
(1.4) Pq(huh3) = l3q{h1,h2) + I3q{h2,h3) V / i , G H 3 .
Sometimes the Busemann cocycle is defined with the opposite
sign, e.g., see [KaOOa].
1.6. DEFINITION. The vertical 1-form
u\ =dpq(hu-)
is the differential of the Busemann cocycle (5q with respect to the second argument
(due to the chain rule (1.4) it is independent of the choice of the first argument
hi), so that
(1.7) Pq(h1,h2)= I u\,
for any smooth path $ with endpoints hi,h2. The vertical vector field v^ is dual
to the form UJ^ with respect to the hyperbolic metric and consists of unit length
vectors "pointing" at the point q in the sense that the integral curves of the field
v]j are the vertical geodesies which converge to the point q at +00 in the visibility
topology. The vertical flow
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