Group theory Research
Recent E-prints:
(For selected pre-2000 papers
click here.)
-
(with T. R. Riley) Some duality conjectures
for finite graphs and their group theoretic
consequences [1 March 2003; revised 11 August 2004,
to appear in Proc. Edin. Math. Soc.]:
.pdf.
This is the second of 3 projected papers on filling
length. It concerns statements of the graph theoretic
Conjecture H and its group theoretic consequences.
The ``electrostatic model" for central extensions is
introduced and it is shown how it enables one to
estimate filling functions with the aid of Conjecture
H. Various other filling functions are introduced
and are shown to agree with filling length up to
equivalence, under the assumption of Conjecture H.
Finally we examine consequences of uniformly bounded
vertex valences in van Kampen diagrams on filling
functions, which is the analog in geometry
of curvature bounded away
from negative infinity.
The third paper in the series will be devoted to
the proof of Conjecture H.
-
(with T. R. Riley) The gallery length filling
function and a geometric inequality for
filling length [25 January 2003, revised 27 May 2005],
accepted for publication in Proc. London. Math. Soc.:
.pdf
This is the first of 3 projected papers on filling
length in group theory, as the analogue of "space"
in computer science, its calculation using duality of
planar graphs, and its relation with isoperimetric
inequalities for central extensions. The first
paper contains a new proof of the double exponential
theorem based on duality, using estimates of the new
filling function, gallery length, introduced here.
-
(with D. F. Holt and T. R. Riley) Isoperimetric inequalities
for nilpotent groups
.pdf
(appeared in GAFA,
Geom. Funct. Anal. 13 (2003), no. 4, 795--814).
This paper contains the first complete proof of the "c+1"
theorem, that states that every finitely generated nilpotent
group of nilpotence class c has an isoperimetric polynomial
of degree c+1.
- (with T. R. Riley) Filling radii of finitely presented
groups
.pdf
(appeared in Quart. J. Math., Oxford,
53 No. 1 (2002) 31-45).
- (with T. R. Riley) Filling length in
finitely presentable groups
.pdf
(appeared in Geom. Ded. 92 No. 1 (2002) 41-58).
This paper contains fundamental results about the filling
length, which plays the same role to "space" in
computer science that area plays to "time". It is
shown that the filling length is bounded by the product
of the filling diameter and the logarithm of the filling
area, where both are calaculated on the same van Kampen
fillings.
- (with H. B. Short) Some isoperimetric inequalities
for kernels of free extensions
.pdf
(appeared in Geom. Ded. 92 No. 1(2002) 63-72).
A finitely presented group which occurs as the kernel
of a homomorphism of a hyperbolic group to a free
group satisfies a polynomial isoperimetric
inequality.