Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 17.05.2002, 16:00--17:45, room 311

P.Svetlov (SPb). 
"Difference equations on graphs and geometry of 3-manifolds"

According to Thurston's Geometrization Conjecture, almost every 
3-manifold can be pasted from Seifert and hyperbolic ``pieces"
(``almost" means all but $Sol$-manifolds).

The main object of our talk is graph manifolds
(3-manifolds which are pasted from only Seifert pieces
with torical boundary along their boundary components).
Let $\Gamma_M$ be a labeled graph which is conjugate to
the splitting of $M$ into Seifert pieces.
It turns out that certain geometrical and topological
properties of $M$ are encrypted in $\Gamma_M$.
For example,  $M$ carries a metric of non-positive
sectional curvature (NPC-metric) iff some second order difference
equation on $\Gamma_M$ has a solution. Investigation of this equation
(which was discovered by S.~Buyalo and V.~Kobel'skii in 1995
and independently by W.~D.~Neumann in 1998) leads to the following

Theorem. Let $M$   be a compact graph manifold.
If $M$ admits an NPC-metric then $M$ can be
finitely covered by a surface bundle over the circle.

In our talk we present a survey of properties of compact
graph manifolds which can be described in terms
of the equation we mentioned above.
The list of such properties includes
   existence of embedded (immersed)
incompressible surface of genus $g\ge 2$ in $M$,
   existence of such surface in some
finite covering space,
   existence of codimension one fibration on $M$.

Authors of the mentioned results:
S.~V.~Buyalo, V.~L.~Kobelskii, W.~D~.Neumann,
J.~Luecke, Y.~Wu, S.~Wang, H.~Rubinstein, and the speaker.