Graphs, free groups and the Hanna Neumann conjecture
A new bound for the rank of the intersection of finitely generated subgroups
of a free group is given, formulated in topological terms, and
very much in the spirit of Stallings. The bound is a
contribution to (although unfortunately not a solution of) the strengthened
Hanna Neumann conjecture.
Journal of group theory (to appear), arXiv.math.GR/0701214.
Partial mirror symmetry I: reflection monoids (with John Fountain)
This is the first of a series of papers in which we initiate and develop the
theory of reflection monoids, motivated by the theory of reflection groups.
The main results identify a number of important inverse semigroups as reflection
monoids, introduce new examples, and determine their orders.
arXiv.math.GR/0701313.
Galois theory, graphs and free groups
A self-contained exposition is given of the topological and Galois-theoretic
properties of the category of combinatorial 1-complexes, or graphs,
very much in the spirit of Stallings.
A number of classical, as well as some new results about free groups are
derived.
arXiv.math.GR/0606326.
(original article)
The smallest hyperbolic 6-manifolds (with John Ratcliffe and Steve Tschantz)
By gluing together copies of an all-right angled Coxeter polytope
a number of open hyperbolic 6-manifolds
with Euler characteristic -1 are constructed. They are the first known
examples of hyperbolic 6-manifolds having the smallest possible volume.
Electronic Research Announcements of the American Mathematical Society, 11 (2005), 40-46.
(original article)
Coxeter groups and hyperbolic manifolds (21 pages)
The rich theory of Coxeter groups is used to provide an algebraic construction of finite
volume hyperbolic
n-manifolds. Combinatorial properties of finite images of these groups can be used to
compute the volumes of the
resulting manifolds. Three examples, in 4,5 and 6-dimensions, are
given, each of very small volume, and in one case of smallest possible
volume.
Mathematische Annalen
, 330 (2004), 127-150.
(original article)
3-Manifolds
from Platonic Solids (8 pages)
The problem of classifying the orientable spherical, Euclidean and
hyperbolic 3-manifolds that arise by identifying the faces of a regular
solid is phrased in the language of Coxeter groups. This allows us to complete
the classification begun in the compact case by Richardson, Rubinstein
and Lorimer.
Topology and its Applications, 138 (2004), 253-263.
The Geometry and Topology of Groups (60 pages).These notes are currently being substantially revised. The posted ones are an older
version. Check with me for details about the new version.
Notes based on a series of lectures given at the Universidad Autonama Madrid in
the Spring of 2003 and the University of York in the Autumn/Spring of 2001-2002
(original
article)
Alternating Quotients of Fuchsian Groups (20 pages).
It is shown that any Fuchsian group has among its homomorphic images
all but finitely many of the alternating groups A_n. This settles in the
affirmative a long-standing conjecture of Graham Higman.
Journal of Algebra 223 (2000) 457-476.
Alternating Quotients of Fuchsian Groups Preprint version (25 pages).
The previous paper refers to the preprint version for a few isolated cases in the
argument.
Constructing
Hyperbolic Manifolds (8 pages) (with Colin Maclachlan)
In this paper we show how to obtain representations of Coxeter groups
acting on H^n to certain classical groups G. We determine when the kernel
K of such a homomorphism is torsion-free and thus H^n/K is a hyperbolic
n-manifold. As an example, this is applied to the two groups described
above, with G suitably interpreted as a classical group. Using this, further
information on the quotient manifold is obtained.
in Computational and Geometric aspects of Modern Algebra, Michael
Atkinson et al. (Editors) London Math. Soc. Lect. Notes,
275 (2000)
78-86. Cambridge University Press.
The Dimension of Varieties over Groups (with Guinevere Dyker) under construction
(original article)
A
Family of Conformally Asymmetric Riemann Surfaces (4 pages)
We give explicit examples of asymmetric Riemann surfaces (that is,
Riemann surfaces with trivial conformal automorphism group) for all genera
greater than or equal to 3. The technique uses Schreier coset diagrams
to construct torsion-free subgroups in groups of signature (0;2,3,r) for
certain values of r.
Glasgow Mathematical Journal 39 (1997), 221-225.
Alternating Quotients of the (3,q,r) Triangle Groups (16 pages) (no pdf version)
A long standing conjecture (attributed to Graham Higman) asserts
that each of the (p,q,r) triangle groups for 1/p+1/q+1/r<1 contains
among its homomorphic images all but finitely many of the alternating or
symmetric groups. This phenomenon has been termed property $\cal H$ by
Mushtaq and Servatius. The work of several authors over the last decade
and a half has shown that for any value of q, there are only finitely many
r such that the (2,q,r) triangle group fails to have property $\cal H$.
In this paper, the techniques used by these authors are generalised to
handle the possibility that p is odd, and as a result, it is shown that
for any q greater than or equal to 3, there are only finitely many r such
that the (3,q,r) triangle group fails to have property $\cal H$.
Communications in Algebra 25 (1997), 1817-1832.
(original article)
Regular Maps on Non-orientable Surfaces (10 pages) (with Marston Conder)
It is well known that regular maps exist on the projective plane but not on the
Klein bottle, nor the non-orientable surface of genus 3. In this paper several infinite families
of regular maps are constructed to show that such maps exist on non-orientable surfaces of over
77 per cent of all possible genera
Geometriae Dedicata 56 (1995), 209-219.
Sydney, Australia; October 2004.
Austin, Texas; October 2003. Algebraic approaches to the construction of hyperbolic
manifolds, particularly arising from representations of Coxeter groups (coming soon).
Madrid, Spain; April 2003. A series of lectures, entitled
The Geometry and Topology of Groups.
(see above)
St Petersburg, Russia; September 2002. The congruence subgroup problem, in particular
Serre's conjecture for arithmetic lattices in the real rank 1 simple Lie groups.
Trento, Italy; July 2001. Constructing geometric manifolds and an earlier talk than the above
on the congruence subgroup problem.
Reviews
of the above and
Reviews
written
Rigidity and arithmeticity, notes for a series of lectures on the rigidity
and arithmeticity of lattices in semi-simple Lie groups (under revision).
Primer on elementary commutative algebra, very elementary (and short!) but mainly to help
understand the motivation behind the concepts introduced in the two papers at the bottom.
Notes on the simple groups of Lie type, (under revision).
back to Brent
Everitt