Combinatorics and geometry in hyperbolic 3-manifolds
To a surface S of finite topological type we may attach the following
objects:
-
The Teichmüller space T(S).
-
The set H(S) of complete hyperbolic structures on the 3-manifold
S
x
R.
-
The geodesic laminations G(S).
-
The "complex of curves" C(S).
I will discuss some relationships between these objects. In particular,
every element N in H(S) determines elements in T(S)
and G(S) called its end invariants (this is a slight simplification)
which describe the asymptotic geometry of the ends of S x R.
A fundamental conjecture of Thurston asserts that these invariants determine
N
completely. C(S) is a simplicial complex encoding the intersection
properties of simple curves in S. It has a natural boundary at infinity
which lies in G(S), and I will discuss the relationship of
this with the geometric properties of hyperbolic manifolds N in
H(S).
In particular there is a criterion, depending only on end invariants, for
N to have "bounded geometry", and more generally a recipe for determining
exactly the set of short geodesics in N, based on quantities defined
from its end invariants as viewed from C(S). There are consequences
of this (some proven, others hoped for) for the classification problem
for hyperbolic 3-manifolds.
Spring 2001 meeting of the PNGS