[GAP Forum] van Kampen diagrams

[Petra Holmes]

Dear Group Pub Forum,

I have been told to send this to everyone I can think of.  Any ideas on it 
are welcome.

Beth

---------- Forwarded message ----------
Date: Mon, 8 Jan 2007 08:26:37 +0000
From: Richard Parker <RParker at amadeuscapital.com>
To: Beth Holmes <P.E.Holmes at dpmms.cam.ac.uk>
Subject: The central problem


Let P be a polyhedral ball which topologically is a sphere,
and where precisely three faces meet at every vertex.

For each face with n sides, define the "curvature" of that
face to be 6-n, so a pentagon has curvature 1, an octagon
has curvature -2 and so on.  The faces do not need to be
regular.

Euler's formula, F + V = E + 2, implies that the total curvature
over the whole of P is 12.

A "patch" of f faces is a subset of the faces of P that has
no holes in it.  Technically it is simply connected.

I need to prove or disprove the following result.

For every such P there is a sequence of patches, each containing
one more face than the previous, such that the sum of the curvature
of all the faces in each patch is greater than zero.  In other words
we can build P one face at a time such that the curvature of the
patch we have made so far is always positive.

This result has huge implications for an algorithm for finitely
presented groups.  P is a van Kampen diagram, and I want to know
whether a short relator with a long proof can be built up relator
by relator looking only at sensible things along the way.