We'll discuss pants (thrice punctured spheres) in
hyperbolic 3-manifolds.

We show that the simple loop conjecture holds for
pants. A proper map of a pants to a hyperbolic
3-manifold which is essential on embedded
simple closed curves and arcs is pi_1-injective
(in fact, by showing that it is totally geodesic).

We show that embedded pants are detected by ideal
points of the character variety.

We show that if a hyperbolic 3-manifold contains
an immersed pants, then it is obtained by Dehn
filling on one component of the Whitehead link
complement, and the immersed pants has a single
clasp, i.e. a single arc of double points.

We'll also describe related results of Shawn
Rafalski about immersed turnovers. An immersed
turnover in a hyperbolic orbifold
is contained in a canonical ``turnover core",
which is an embedded suborbifold with turnover
boundary and bounded volume. This is an analogue
of the JSJ decomposition for 3-manifolds.

Lecture #12605

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